T-systems and Y-systems in integrable systems

T-systems and Y-systems in in tegrable systems A tsuo Kuniba a , T omoki Nak anishi b and Jun ji Suzuki c De dic ate d to the memory of Pr ofessor Morikazu T o da a Institute of Physics, Univ ersity of T okyo, Komaba, T okyo, 153-89 02, Japan b Graduate School of Mathematics , Nag oya Univ e r sity , Nag oy a, 464-860 4, Japan c Department of Physics, F acult y of Science, Shizuok a Universit y , Ohy a, 836, Japan Abstract T and Y-systems are ubiquitous structures in clas sical and quantum in tegr able sys- tems. They are diﬀerence equa tions having a v ar iety of asp ects rela ted to comm ut- ing transfer matrice s in so lv able lattice mo dels, q -character s of Kirillov-Reshetikhin mo dules of qua ntum aﬃne algebr as, cluster a lgebras with co eﬃcients, pe r io dicity conjectures of Z a molo dchik ov a nd others , dilog arithm identities in confor mal ﬁeld theory , diﬀerence analo g o f L -o p e rators in KP hiera r ch y , Stokes phenomena in 1D Schr¨ odinger problem, AdS/CFT corres po ndence, T o da ﬁeld equations o n discr ete spacetime, Laplace sequence in discr ete geometry , F ermionic character form ulas and combinatorial completenes s of Bethe ansatz, Q -system a nd ideal gas with exclusio n statistics, analytic a nd ther mo dynamic Bethe ans¨ a tze , quantum transfer matrix metho d and so forth. This re v iew article is a collectio n of short rev iews on these topics which can b e re a d mor e or less indepe nden tly . 1 2 Contents 1. Int ro duction 4 1.1. T and Y-systems 4 1.2. Cont ent s and brief guide 5 2. T and Y-systems for quantum a ﬃne algebras and Y angians 8 2.1. Un twisted case 8 2.2. Restriction 13 2.3. Relation b etw een T and Y-systems 13 2.4. Twisted case 15 2.5. Restriction and relations b etw een T and Y-systems 18 2.6. U q ( sl ( r | s )) c ase 18 2.7. Bibliographica l notes 20 3. T-system among commutin g transfer matrices 21 3.1. V ertex mo dels and fusion 21 3.2. T ransfer matrices 23 3.3. Restricted solid-on- solid (RSOS) models and fusio n 25 3.4. Relation to vertex mo dels 27 3.5. Restriction 28 3.6. T ransfer matrices 29 3.7. V ertex and RSOS mo dels for gener al g 30 3.8. Bibliographica l notes 33 4. T-system in quantum gr oup theory 34 4.1. Quantum aﬃne alge br a 34 4.2. Finite dimensional repre s entations 35 4.3. Example 35 4.4. q -characters 37 4.5. T-system and q -characters 40 4.6. T-system for quantum aﬃniza tions of quant um Kac-Mo o dy algebra s 41 4.7. Bibliographica l notes 45 5. F ormulation by cluster algebra s 45 5.1. Dilogarithm ident ities in conforma l ﬁeld theo ry 45 5.2. Cluster algebr a s with co eﬃcients 47 5.3. T and Y-systems in cluster alg ebras 49 5.4. Application to p erio dicity a nd dilogarithm ident ities 52 5.5. Bibliographica l notes 52 6. Jacobi-T rudi type for m ula 53 6.1. In tr o duction: Type A r 53 6.2. T y pe B r 54 6.3. T y pe C r 55 6.4. T y pe D r 56 6.5. Another Jacobi- T rudi type for mula for B r 57 6.6. Bibliographica l notes 58 7. T ableau sum for mula 58 7.1. T y pe A r . 58 7.2. T y pe B r 59 7.3. T y pe C r 60 7.4. T y pe D r 61 7.5. Bibliographica l notes 62 3 8. Analytic Bethe ansatz 62 8.1. A 1 case 63 8.2. Dressed v acuum form and q -characters 65 8.3. RSOS mo dels 68 8.4. Bibliographica l notes 69 9. W ronskia n type (Ca soratian) formula 69 9.1. Diﬀerence L op erato rs 69 9.2. Casoratia n formula 70 9.3. Q -functions 72 9.4. B¨ acklund transformations 73 9.5. T y pe C r 73 9.6. T y pe B r and D r 75 9.7. T y pe sl ( r | s ) 76 9.8. Bibliographica l notes 77 10. T-system in ODE 77 10.1. Generalized Stokes multipliers - the 2nd order case 78 10.2. Higher order ODE 79 10.3. W ronskia n-Casor atian duality 81 10.4. Bibliogra phica l no tes 82 11. Applications in gaug e /string theo ries 82 11.1. Planar AdS/CFT sp ectrum 83 11.2. T and Y-system for AdS 5 /CFT 4 83 11.3. F ormula for planar AdS/CFT sp ectrum 84 11.4. Asymptotic Bethe ansatz 85 11.5. Area of minimal surface in AdS 86 11.6. Stokes phenomena, T a nd Y-system 88 11.7. Asymptotics, WKB and TBA 88 11.8. Area and free energy 90 11.9. Bibliogra phica l no tes 91 12. Aspects as classical integrable system 91 12.1. Contin uum limit 92 12.2. Discrete geometry 95 12.3. Bibliogra phica l no tes 97 13. Q-system and F ermionic for mula 98 13.1. Int ro duction 98 13.2. Simplest example of M λ 98 13.3. Simplest example of N λ 100 13.4. Theorems for t y pe A 1 101 13.5. Multiv ariable Lagra nge inv er sion 104 13.6. Q-system and theorems for g 107 13.7. Q ( a ) m as a classical character 110 13.8. Bibliogra phica l no tes and further asp ects 111 14. Y-system and thermo dynamic Bethe ansa tz 112 14.1. Y-system for ADE and deformed Carta n matr ic e s 112 14.2. TBA kernels 114 14.3. Y-system for g from TBA equation 115 14.4. Constant Y-sys tem 116 14.5. Relation with Q-system. 11 9 4 14.6. Q ( a ) m at ro ot of unit y 120 14.7. Bibliogra phica l no tes 12 1 15. TBA analysis of RSOS mo dels 122 15.1. TBA equation 122 15.2. High temp e rature entrop y 125 15.3. Cent ral charges 126 16. T-system in use 129 16.1. Correla tion leng ths of vertex models 129 16.2. Finite size correctio ns 132 16.3. Quantum transfer matr ix approach 137 16.4. Simpliﬁed TBA equations 141 16.5. Hybrid equations 143 Ac knowledgmen ts 146 References 146 1. Introduction 1.1. T and Y- s ystems. The T-system is a diﬀerence equa tion a mong commuting v ariables T ( a ) m ( u ), most typically app ea r ing as ( m ∈ Z ≥ 0 ) T ( a ) m ( u − 1) T ( a ) m ( u + 1) = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) + T ( a − 1) m ( u ) T ( a +1) m ( u ) . Originally it was found as a functional rela tio n in 2D solv able lattice mo dels in statistical mechanics [1]. In this cont ext, T ( a ) m ( u ) is a c ommut ing row tra nsfer matrix in the sense of Baxter [2 ] la be led with ( a, m ) and having the sp ectr al par a meter u 1 . The Y-system is another diﬀerence equa tion, t ypic a lly like ( m ∈ Z ≥ 1 ) Y ( a ) m ( u − 1) Y ( a ) m ( u + 1) = (1 + Y ( a − 1) m ( u ))(1 + Y ( a +1) m ( u )) (1 + Y ( a ) m − 1 ( u ) − 1 )(1 + Y ( a ) m +1 ( u ) − 1 ) . It w as extracted as a univ er sal functional r elation in thermo dynamic Bethe a ns atz (TBA) for solv able lattice mode ls a s well as (1 + 1)D int egrable quantum ﬁeld theory mo dels [3, 4 , 5]. In this con text, Y ( a ) m ( u ) stands for the B oltzmann factor of an excitation mo de in the sense of Y ang-Y ang [6 ] lab eled with ( a, m ) and having the rapidity u . As such, the b o th sys tems o riginate in Y ang-B axter quantum integrable systems but are apparently c o ncerned with the ob jects that are not related to o directly . The ﬁrst curiosity is nevertheless that the formal substitution Y ( a ) m ( u ) = T ( a − 1) m ( u ) T ( a +1) m ( u ) T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) provides a solution to the Y-system in ter ms of the T-system. Moreov er, such a canonica l pa ir of companion sy stems can b e for m ulated uniformly for all the classical simple Lie alg e br as g [1] 2 . Now we ca n give a deferred expla nation of the sup e rscript a ; it runs ov er the vertices of the Dynkin diag ram of g . The ab ove 1 By T we m ean t T ransfer matrices, but it can either b e though t as T oda or T au. 2 Actually to b e understo o d as Y angian Y ( g ) or un twisted quant um aﬃne algebra U q ( ˆ g ). Twisted case is also kno wn. See Remark 2. 1. 5 formulas are just the examples from t yp e A 3 , where the cas e g = A 1 go es back to [7]. In the r elev ant developmen ts a cross the centuries, the T and Y-systems have turned o ut to be ubiquitous s tructures with a wealth of applications. F or instance, they emerg e in q -characters for Kirillov-Reshetikhin mo dules of qua nt um aﬃne algebras , exchange relations in cluster algebr a s with co eﬃcients, p erio dicity conjec- tures of Za molo dchik ov and o ther s, diloga rithm identities in conformal ﬁeld theo ry (CFT) and their functional genera lizations, dres sed v acuum for ms in analytic Bethe ansatz, Stokes pheno mena in ordinary diﬀerential equa tions, anomalous scaling di- mensions o f N = 4 sup er Y ang-Mills oper a tors, area of minimal surface in AdS, Laplace se q uence of quadrilateral lattice in discre te geometry , tau functions in lat- tice T o da ﬁeld equations, F ermionic formulas for br anching co eﬃcients and weigh t m ultiplicities for Lie algebra characters , combinatorial completeness o f string hy- po thesis in Bethe ansatz, Q -system and gra nd partition function o f ideal g as with exclusion statistics, quantum tra nsfer matrix approach to ﬁnite temp erature prob- lems and so on. This re v iew is a collectio n of brief exp o s itions of these topics where the T and Y-systems hav e play ed key r oles. It co nsists of sectio ns of mo derate length w hich are not to o mutually dep endent. A more detailed account o f the conten ts ca n b e found in Section 1.2. As an ov e r view, T- systems are fundamental structure s reﬂecting s y mmetries and algebraic as pec ts of the pr o blems rather directly . They ca n also accommo da te v arious gauge/ normalizatio n freedom o f co ncrete mo dels. On the other hand, Y- systems a re more universal b eing more or less fre e from such deg rees of freedom. They are suitable fo r pr actical applica tions with a ppropriate ana ly ticit y input. In fact, the connection b etw een the T a nd Y-sys tems men tioned prev iously has o p ened a route to establish TBA type integral equatio ns dir ectly from tra nsfer matr ices without reco urse to the TBA its e lf. In this sense, Y-systems are the forma t in which the symmetries enco ded in the T-systems a re mo s t eﬃciently utilized as a practical implemen t. In the light of ever gr owing p ersp ectives, what sort of eq uations or structur es a re to be r ecognized as T or Y-systems is actually a matter of time-dep endent option. F or instance from a n algebra ic p oint of view (leaving analytic asp ects), T-s ystems hav e b een gener a lized broadly to the quantum aﬃnization of q uantum Kac- Mo o dy algebras b y Her nandez [8] (Section 4 .6). Cluster algebra with co eﬃcients by F omin and Zelev insky [9] oﬀers a comprehensive scheme to generalize and control the T and Y-systems simult aneously b y quiv er s (Section 5). Nonetheless, this paper is mostly devoted to the description of basic results concerning the aforementioned “classic” T and Y-systems a s so ciated with g . W e ther e fore lo ok forward to the nex t review to come, hop efully someday by some author, bringing a delightf ul renewal. 1.2. Con tents and brief gu i de. Here ar e abstracts o f the subsequen t s ections. They will be follow ed b y another brief guide to the pap er . Section 2 . The T and Y-s y stems for unt wis ted and t wisted qua nt um aﬃne alge- bras ar e pr e s ented. They hav e unre s tricted and level ℓ restr ic ted versions. Those for Y a ngian a re formally the same with the unrestricted ones for the un twisted 3 The T- system for type A f ormally coincides with what is known as the Hir ota-Miwa equation in soliton theory , which was an unexp ected l ink also to classical inte gr able systems. 6 quantum aﬃne a lgebra U q ( ˆ g ), where g denotes a ﬁnite dimensional simple Lie al- gebra thr o ughout the pap er. W e als o include the U q ( sl ( r | s )) case. This section is meant to be the r eference of these systems thr oughout the pap er. The ﬁrst prop- erty , T-system provides a solution to Y-system, is stated. Subsequent sectio ns will mainly be concerned with the unt wisted case U q ( ˆ g ) 4 . Section 3. The T-system was originally disc overed as functiona l relations a mong commuting transfer matrices for so lv able la ttice mo dels in statistical mechanics. W e give an elementary ex po sition o f s uch contexts for the b oth vertex and restricted solid-on-s o lid (RSOS) mo dels along with their fusion pro cedure. The tw o types of mo dels are related to the unrestricted and re stricted T-systems, resp ectively . Section 4. W e describe the background of the T- system in the representation theory of q uantum aﬃne alg e br as such as classiﬁcation of irreducible ﬁnite dimen- sional re presentations, Kirillov-Reshetikhin mo dules and q -character s . The funda- men tal results ar e that q -characters of the Kirillov-Reshetikhin mo dules satisfy the T-system (Theorem 4.8) a nd the des cription of the Gr othendieck ring Rep U q ( ˆ g ) by the T-sys tem (Theore m 4 .9). A bro a d extension of the T-system to the qua nt um aﬃnization of quantum K a c-Mo o dy a lgebras is also men tione d. The results o f this section are not necessary elsewhere except the basics of q -characters whic h will b e men tioned in table a u sum for mulas (Section 7 ), a nalytic Bethe ansatz (Section 8) and Q-system (Section 13). Section 5. The cluster a lgebra with coeﬃcients is built up on cluster v ar iables and co eﬃcient tuples ob eying cer tain exchange relatio ns controlled by a quiver. W e demonstrate how such a setup enco des the T and Y-systems simultaneously in an essential wa y . It op ens a fr uitful link with the cluster category theo ry , which led to a ﬁnal pro of of the dilogar ithm ide ntities in confor mal ﬁeld theory a nd the per io dicity conjecture on the b oth sys tems for arbitra ry level and g . Section 6. J acobi-T r udi type deter minant formulas are listed for T-sys tems for non exceptional g . The type C r and D r cases in volve Pfaﬃans as well. Section 7. T ableau sum formulas are presented for T-systems for no n exceptio nal g along the cont ext of q -characters. Section 8. W e argue the rela tion b etw een q -characters a nd eigenv alue formulas (dressed v acuum forms) of tr a nsfer matrice s in solv able la ttice mo dels by analytic Bethe ans a tz. Combined with the results in Section 7, it leads to solutions of T- systems in ter ms o f the Baxter Q- functions . W e ma inly concern vertex mo dels a nd include a brief argument on RSOS mo dels . Section 9. W e intro duce a diﬀer ence analog of L -op erato r s in solito n theory to construct solutions to the T-systems for g = A r and C r by Casor atians (diﬀerence analog of W ronsk ians). The B a xter Q-functions ar e identiﬁed with a sp e cial cla ss of Ca s oratians and gener alized to a wider family of functions that admit B¨ a cklund transformatio ns. Analogous diﬀerence L -op era tors a re presented also for B r , D r and sl ( r | s ). Section 10. A restr icted T-system for A 1 emerges in Stok es phenomena o f 1 D Schr¨ odinger equation with a speciﬁc potential. Similar facts hold also for the T- system for A r and a class o f ( r + 1)th order o rdinary diﬀerential equation (ODE). W ronskia ns for these equations ev aluated at the or igin play an analogous ro le to the Ca soratians in Section 9 (W r onskian-Ca soratian duality). W e describe these 4 Th us i n most si tuations we will say simply T and Y-systems f or g instead of U q ( ˆ g ). 7 features that sta y within an elemen tar y algebraic par t in the so- called ODE /IM (in tegrable mode ls ) corres po ndence. Section 11. This s e c tion is most hep-th o riented. W e brieﬂy digest a pplications of some sp eciﬁc T and Y-systems in the tw o topics fro m the AdS/CFT corr e sp o ndence. The ﬁrst is from the gauge theory a bo ut the anomalous scaling dimensio ns (planar AdS/CFT sp ectrum) of N = 4 super Y ang- Mills op erato rs. The second is the ar ea of the minimal s urface in AdS from the string theory , which is relev a nt to gluon planar sca ttering amplitudes. The analysis in the latter topic inv o lves the Stokes phenomena re lated to a g eneralized sinh-Go rdon equa tion, w hich may b e viewed a s a generaliza tion of the ODE/ IM corresp ondence mentioned in Section 1 0. Section 12. Contin uous limits o f the T-system for g yie ld the diﬀerence-diﬀerential or 2 D diﬀere n tial equations known as the (lattice) T oda ﬁeld equation. Their Ha mil- tonian structur e is pre sented for ge ne r al g . W e a lso discuss an asp ect from classica l discrete g eometry , where the Y-system for A ∞ arises as the Laplace sequence of quadrilatera l lattice, the discr ete geometry analog of the conjugate net. Section 13. T-system without sp ectr al para meter is called Q - system 5 . W e sys- tematically constr uct certain p ower ser ies s olutions to the (generalize d) Q-sy stem by multi-v a riable Lag range inv ersio n. As a co rollar y of this and results fro m Section 4, the so-c alled F ermio nic ch aracter formula for the Kirillov-Reshetikhin mo dules is fully established for a ll g . Physically , this problem is also connected to the grand partition function of ideal gas with exclusion statistics. These results ar e re v iewed in conjunction with the intimately related sub ject known as combinatorial co m- pleteness of Bethe ansatz for U q ( ˆ g ) b oth at q = 1 and q = 0, where the c a se q = 1 go es back to Bethe [10], the go dfa ther o f the sub ject, himself. Section 1 4. W e expla in how the Y-s ystem for g emerges fro m the TBA eq uation asso ciated to U q ( ˆ g ) with q b eing a ro ot of unity deriv ed in Section 15. V arious relations among the TBA kernels a re summar ized. The constant Y- s ystem is intro- duced a nd rela ted to the Q-system. They ar e essential ingredients in the dilog arithm ident it y (Section 5.1) a nd the TBA analysis o f RSOS mo dels (Section 15). As a related issue , w e brieﬂy discuss the Q-s y stem at ro o t o f unity including Conjecture 14.2. Section 15. The U q ( ˆ g ) Bethe equa tion with q a ro ot of unit y is r elev ant to the critical RSO S mo dels sketc hed in Sectio n 3.3. W e outline the TBA analys is to ev aluate the high temperature entropy by the level restricted Q-system (Section 14.5 – 14.6) and central charges by the dilog arithm identit y (Sectio n 5.1). The TBA equation obtained here uniformly for gener al g is the orig in of our Y-system as shown in Section 14 .1 and 14 .3. Section 16. T he ﬁnite size or ﬁnite temp e rature problems in solv able la ttice mo dels are a nalyzed eﬃciently by the use of T a nd Y-systems without relying on TBA approach a nd string hypo thesis. W e illustrate v ar ious suc h methods a long the simplest vertex and RSOS mo dels ba s ed on g = A 1 . W e a lso include a simple application of the per io dicity of the le vel 0 restricted T-sy stem to the calc ula tion of correlatio n lengths of vertex mo dels in Section 16.1. Let us close the in tro duction with yet another brief guide of the conten ts. As we already ment ioned, Sectio n 2 is the collection of the basic data; concrete forms of the T a nd Y-systems that will be considered in the review and deﬁnitions/notations concerning the ro ot s ystem of g . With re g ard to the subs equent sections, it is 5 This Q i s unrelated with Baxter’s Q-f unctions. See Section 13.8 for the or igin of the name. 8 to o demanding to as sume the familiarity of the conten ts in earlier sections. So we hav e avoided such a style a nd tried to make ea ch sectio n into a more or less independently readable rev iew on a sp eciﬁc topic ar ound ten pages . Most of them contain bibliog raphical notes at the end, which hop efully help the readers ga in more per sp ectives into the sub jects and a ctivities around. There are nevertheless several sections that are intimately rela ted or par tly de- pendent of course. Roughly , t hey may b e gr oup ed (no n exclusively) under the following theme. • Solv able lattice mo dels and their analysis : Sections 3, 8, 15, 16. • Kirillov-Reshetikhin mo dules and their q -characters: Sections 4, 7, 8, 13. • V ariety o f solutions to the T-system: Sections 6, 7, 8, 9. • Stokes phenomena: Sectio ns 10, 11. • Q-system and constant Y-sy s tem: Sections 13, 14. • Y-system and TBA: Sections 11, 14, 15. 2. T and Y -systems for quantum affine al gebras and Y angians W e present the T-system a nd Y-sys tem asso ciated with unt wisted and twisted quantum aﬃne algebras. They have unres tricted and lev el restricted versions. Those for Y angian are formally the same with the unrestricted ones for the un- t wisted q ua ntu m aﬃne algebra s. W e also include the case U q ( sl ( r | s )). This section is devoted to the presentation of these systems with the basic data on ro o t systems. Thu s we will only state their ﬁrst pro p erty , T-sy stem provides a solution to Y- system, in Theor em 2.5. leaving the exp osition of v ariety of asp ects in subsequent sections. 2.1. Un twisted case. Let g b e a simple Lie alg ebra ass o ciated with a Dynkin diagram of ﬁnite type. W e set I = { 1 , . . . , r } with r = rank g and en umerate the vertices of the Dynkin dia g rams as Figure 1. W e follow [1 1] exc ept for E 6 , for which we choos e the one natura lly corr esp onding to the enumeration of the t wisted aﬃne diagram E (2) 6 in Section 2.4. With a s light a buse of nota tio n, we will write for example g = A r to mean that g is the one as s o ciated with the Dynkin diagra m of t yp e A r . The cas es A r , D r , E 6 , E 7 and E 8 are referre d to a s simply laced. W e set num b ers t a nd t a ( a ∈ I ) by t =      1 g : simply lace d , 2 g = B r , C r , F 4 , 3 g = G 2 , t a =      1 g : simply laced , 1 g : no ns imply laced, α a : long ro o t , t g : nonsimply laced, α a : shor t roo t . (2.1) Let α a , ω a ( a ∈ I ) b e the simple ro ots a nd the fundamental weigh ts of g . W e ﬁx a bilinear form ( | ) o n the dual space of the Cartan subalgebr a no rmalized as ( α a | α a ) = 2 t a , ( α a | ω b ) = δ ab t a . (2.2) Let C = ( C ab ), C ab = 2( α a | α b ) / ( α a | α a ), b e the Cartan matrix of g . W e have C ab = t a ( α a | α b ), α a = P r b =1 C ba ω b and ( C − 1 ) ab = t a ( ω a | ω b ). W e denote by h and h ∨ the Coxeter num b er a nd the dual Coxeter num b er of g , resp ectively . They are 9 listed as follows with the dimension of g . g A r B r C r D r E 6 E 7 E 8 F 4 G 2 dim g r ( r + 2) r (2 r + 1) r (2 r + 1 ) r (2 r − 1) 78 133 248 5 2 14 h r + 1 2 r 2 r 2 r − 2 12 18 30 12 6 h ∨ r + 1 2 r − 1 r + 1 2 r − 2 12 18 30 9 4 (2.3) The relation dim g = (1 + h )ra nk g holds as is well known. A r 1 2 r − 1 r B r 1 2 r − 1 r C r 1 2 r − 1 r D r 1 2 r − 1 r − 2 r E 6 1 2 3 5 6 4 E 7 1 2 3 4 5 6 7 E 8 1 2 3 4 5 6 7 8 F 4 1 2 3 4 G 2 1 2 Figure 1. The Dynkin diag rams for g and their enumerations. The unrestricted T-sy s tem for g is the following relations among the commuting v ariables { T ( a ) m ( u ) | a ∈ I , m ∈ Z ≥ 1 , u ∈ U } , where T (0) m ( u ) = T ( a ) 0 ( u ) = 1 if they o ccur in the RHS. F or simply lac e d g , T ( a ) m ( u − 1) T ( a ) m ( u + 1) = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) + Y b ∈ I : C ab = − 1 T ( b ) m ( u ) . (2.4) F or example in type A r , it has the form T ( a ) m ( u − 1) T ( a ) m ( u + 1) = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) + T ( a − 1) m ( u ) T ( a +1) m ( u ) , (2.5) for 1 ≤ a ≤ r with T ( r +1) m ( u ) = 1 . In particular , for A 1 it reads T m ( u − 1) T m ( u + 1) = T m − 1 ( u ) T m +1 ( u ) + 1 (2.6) with the simpliﬁed notation T m ( u ) = T (1) m ( u ). 10 F or g = B r , T ( a ) m ( u − 1) T ( a ) m ( u + 1) = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) (2.7) + T ( a − 1) m ( u ) T ( a +1) m ( u ) (1 ≤ a ≤ r − 2) , T ( r − 1) m ( u − 1) T ( r − 1) m ( u + 1) = T ( r − 1) m − 1 ( u ) T ( r − 1) m +1 ( u ) + T ( r − 2) m ( u ) T ( r ) 2 m ( u ) , T ( r ) 2 m  u − 1 2  T ( r ) 2 m  u + 1 2  = T ( r ) 2 m − 1 ( u ) T ( r ) 2 m +1 ( u ) + T ( r − 1) m  u − 1 2  T ( r − 1) m  u + 1 2  , T ( r ) 2 m +1  u − 1 2  T ( r ) 2 m +1  u + 1 2  = T ( r ) 2 m ( u ) T ( r ) 2 m +2 ( u ) + T ( r − 1) m ( u ) T ( r − 1) m +1 ( u ) . F or g = C r , T ( a ) m  u − 1 2  T ( a ) m  u + 1 2  = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) (2.8 ) + T ( a − 1) m ( u ) T ( a +1) m ( u ) (1 ≤ a ≤ r − 2) , T ( r − 1) 2 m  u − 1 2  T ( r − 1) 2 m  u + 1 2  = T ( r − 1) 2 m − 1 ( u ) T ( r − 1) 2 m +1 ( u ) + T ( r − 2) 2 m ( u ) T ( r ) m  u − 1 2  T ( r ) m  u + 1 2  , T ( r − 1) 2 m +1  u − 1 2  T ( r − 1) 2 m +1  u + 1 2  = T ( r − 1) 2 m ( u ) T ( r − 1) 2 m +2 ( u ) + T ( r − 2) 2 m +1 ( u ) T ( r ) m ( u ) T ( r ) m +1 ( u ) , T ( r ) m ( u − 1) T ( r ) m ( u + 1) = T ( r ) m − 1 ( u ) T ( r ) m +1 ( u ) + T ( r − 1) 2 m ( u ) . F or g = F 4 , T (1) m ( u − 1) T (1) m ( u + 1) = T (1) m − 1 ( u ) T (1) m +1 ( u ) + T (2) m ( u ) , (2.9) T (2) m ( u − 1) T (2) m ( u + 1) = T (2) m − 1 ( u ) T (2) m +1 ( u ) + T (1) m ( u ) T (3) 2 m ( u ) , T (3) 2 m  u − 1 2  T (3) 2 m  u + 1 2  = T (3) 2 m − 1 ( u ) T (3) 2 m +1 ( u ) + T (2) m  u − 1 2  T (2) m  u + 1 2  T (4) 2 m ( u ) , T (3) 2 m +1  u − 1 2  T (3) 2 m +1  u + 1 2  = T (3) 2 m ( u ) T (3) 2 m +2 ( u ) + T (2) m ( u ) T (2) m +1 ( u ) T (4) 2 m +1 ( u ) , T (4) m  u − 1 2  T (4) m  u + 1 2  = T (4) m − 1 ( u ) T (4) m +1 ( u ) + T (3) m ( u ) . F or g = G 2 , T (1) m ( u − 1) T (1) m ( u + 1) = T (1) m − 1 ( u ) T (1) m +1 ( u ) + T (2) 3 m ( u ) , (2.10) T (2) 3 m  u − 1 3  T (2) 3 m  u + 1 3  = T (2) 3 m − 1 ( u ) T (2) 3 m +1 ( u ) + T (1) m  u − 2 3  T (1) m ( u ) T (1) m  u + 2 3  , T (2) 3 m +1  u − 1 3  T (2) 3 m +1  u + 1 3  = T (2) 3 m ( u ) T (2) 3 m +2 ( u ) + T (1) m  u − 1 3  T (1) m  u + 1 3  T (1) m +1 ( u ) , T (2) 3 m +2  u − 1 3  T (2) 3 m +2  u + 1 3  = T (2) 3 m +1 ( u ) T (2) 3 m +3 ( u ) + T (1) m ( u ) T (1) m +1  u − 1 3  T (1) m +1  u + 1 3  . 11 W e no te that these r elations are n ot bilinear in g eneral under the bo undary condition stated b efore (2.4). The second ter ms on the RHS can b e o f order 0,1,2 and 3 in T ( a ) m ( u ). The v ar iable u ∈ U is called the sp e ctr al p ar ameter . The set U can be either the complex plane C , o r the c y linder C ξ := C / (2 π √ − 1 /ξ ) Z such that 2 π √ − 1 /ξ 6∈ Q . The c ho ice will not matter serious ly , but reﬂects the underlying algebra . Remark 2.1. In Section 4 we will see that the T- system fo r g is actually asso ciated with the unt wisted quantum aﬃne algebr a U q ( ˆ g ) with q = e ~ when U = C t ~ . The choice U = C corre s po nds to the Y a ngian Y ( g ) in a simila r s e ns e. In this review we will mostly b e co ncerned with the U q ( ˆ g ) cas e . Thus we have simply chosen to say T-system for g rather than T-system for U q ( ˆ g ). The latter terminology is mor e balanced when the twisted case is considered in Section 2 .4. Note tha t the choice U = C ξ eﬀectively impos es an additional p erio dicity T ( a ) m ( u ) = T ( a ) m ( u + 2 π √ − 1 ξ ). By the assumption 2 π √ − 1 /ξ / ∈ Q , this do es not interfere with the T-system. Similar remarks apply to the Y-system in what follows. The unrestricted Y-system for g is the following relatio ns among co mm uting v ariables { Y ( a ) m ( u ) | a ∈ I , m ∈ Z ≥ 1 , u ∈ U } , where Y (0) m ( u ) = Y ( a ) 0 ( u ) − 1 = 0 if they o ccur in the RHS. F or simply lac e d g , Y ( a ) m ( u − 1) Y ( a ) m ( u + 1) = Q b ∈ I : C ab = − 1 (1 + Y ( b ) m ( u )) (1 + Y ( a ) m − 1 ( u ) − 1 )(1 + Y ( a ) m +1 ( u ) − 1 ) . (2.11) F or g = B r , Y ( a ) m ( u − 1) Y ( a ) m ( u + 1) = (1 + Y ( a − 1) m ( u ))(1 + Y ( a +1) m ( u )) (1 + Y ( a ) m − 1 ( u ) − 1 )(1 + Y ( a ) m +1 ( u ) − 1 ) (2.12) (1 ≤ a ≤ r − 2) , Y ( r − 1) m ( u − 1) Y ( r − 1) m ( u + 1) = (1 + Y ( r − 2) m ( u ))(1 + Y ( r ) 2 m − 1 ( u ))(1 + Y ( r ) 2 m +1 ( u )) × (1 + Y ( r ) 2 m  u − 1 2  )(1 + Y ( r ) 2 m  u + 1 2  ) (1 + Y ( r − 1) m − 1 ( u ) − 1 )(1 + Y ( r − 1) m +1 ( u ) − 1 ) , Y ( r ) 2 m  u − 1 2  Y ( r ) 2 m  u + 1 2  = 1 + Y ( r − 1) m ( u ) (1 + Y ( r ) 2 m − 1 ( u ) − 1 )(1 + Y ( r ) 2 m +1 ( u ) − 1 ) , Y ( r ) 2 m +1  u − 1 2  Y ( r ) 2 m +1  u + 1 2  = 1 (1 + Y ( r ) 2 m ( u ) − 1 )(1 + Y ( r ) 2 m +2 ( u ) − 1 ) . F or g = C r , Y ( a ) m  u − 1 2  Y ( a ) m  u + 1 2  = (1 + Y ( a − 1) m ( u ))(1 + Y ( a +1) m ( u )) (1 + Y ( a ) m − 1 ( u ) − 1 )(1 + Y ( a ) m +1 ( u ) − 1 ) (2.13) (1 ≤ a ≤ r − 2) , Y ( r − 1) 2 m  u − 1 2  Y ( r − 1) 2 m  u + 1 2  = (1 + Y ( r − 2) 2 m ( u ))(1 + Y ( r ) m ( u )) (1 + Y ( r − 1) 2 m − 1 ( u ) − 1 )(1 + Y ( r − 1) 2 m +1 ( u ) − 1 ) , 12 Y ( r − 1) 2 m +1  u − 1 2  Y ( r − 1) 2 m +1  u + 1 2  = 1 + Y ( r − 2) 2 m +1 ( u ) (1 + Y ( r − 1) 2 m ( u ) − 1 )(1 + Y ( r − 1) 2 m +2 ( u ) − 1 ) , Y ( r ) m ( u − 1) Y ( r ) m ( u + 1) = (1 + Y ( r − 1) 2 m +1 ( u ))(1 + Y ( r − 1) 2 m − 1 ( u )) × (1 + Y ( r − 1) 2 m  u − 1 2  )(1 + Y ( r − 1) 2 m  u + 1 2  ) (1 + Y ( r ) m − 1 ( u ) − 1 )(1 + Y ( r ) m +1 ( u ) − 1 ) . F or g = F 4 , Y (1) m ( u − 1) Y (1) m ( u + 1) = 1 + Y (2) m ( u ) (1 + Y (1) m − 1 ( u ) − 1 )(1 + Y (1) m +1 ( u ) − 1 ) , (2.14) Y (2) m ( u − 1) Y (2) m ( u + 1) = (1 + Y (1) m ( u ))(1 + Y (3) 2 m − 1 ( u ))(1 + Y (3) 2 m +1 ( u )) × (1 + Y (3) 2 m  u − 1 2  )(1 + Y (3) 2 m  u + 1 2  ) (1 + Y (2) m − 1 ( u ) − 1 )(1 + Y (2) m +1 ( u ) − 1 ) , Y (3) 2 m  u − 1 2  Y (3) 2 m  u + 1 2  = (1 + Y (2) m ( u ))(1 + Y (4) 2 m ( u )) (1 + Y (3) 2 m − 1 ( u ) − 1 )(1 + Y (3) 2 m +1 ( u ) − 1 ) , Y (3) 2 m +1  u − 1 2  Y (3) 2 m +1  u + 1 2  = 1 + Y (4) 2 m +1 ( u ) (1 + Y (3) 2 m ( u ) − 1 )(1 + Y (3) 2 m +2 ( u ) − 1 ) , Y (4) m  u − 1 2  Y (4) m  u + 1 2  = 1 + Y (3) m ( u ) (1 + Y (4) m − 1 ( u ) − 1 )(1 + Y (4) m +1 ( u ) − 1 ) . F or g = G 2 , Y (1) m ( u − 1) Y (1) m ( u + 1) = (1 + Y (2) 3 m − 2 ( u ))(1 + Y (2) 3 m +2 ( u )) × (1 + Y (2) 3 m − 1  u − 1 3  )(1 + Y (2) 3 m − 1  u + 1 3  ) × (1 + Y (2) 3 m +1  u − 1 3  )(1 + Y (2) 3 m +1  u + 1 3  ) × (1 + Y (2) 3 m  u − 2 3  )(1 + Y (2) 3 m  u + 2 3  ) × (1 + Y (2) 3 m ( u )) (1 + Y (1) m − 1 ( u ) − 1 )(1 + Y (1) m +1 ( u ) − 1 ) , (2.15) Y (2) 3 m  u − 1 3  Y (2) 3 m  u + 1 3  = 1 + Y (1) m ( u ) (1 + Y (2) 3 m − 1 ( u ) − 1 )(1 + Y (2) 3 m +1 ( u ) − 1 ) , Y (2) 3 m +1  u − 1 3  Y (2) 3 m +1  u + 1 3  = 1 (1 + Y (2) 3 m ( u ) − 1 )(1 + Y (2) 3 m +2 ( u ) − 1 ) , Y (2) 3 m +2  u − 1 3  Y (2) 3 m +2  u + 1 3  = 1 (1 + Y (2) 3 m +1 ( u ) − 1 )(1 + Y (2) 3 m +3 ( u ) − 1 ) . W e stress that the T and Y-systems for nonsimply la ced g are not just a folding of simply laced cases. W e also remark that T and Y-systems for B 2 and C 2 are equiv alent and trans- formed to ea ch other by T (1) m ( u ) ↔ T (2) m ( u ) and Y (1) m ( u ) ↔ Y (2) m ( u ) reﬂecting the fact B 2 ≃ C 2 . 13 2.2. Restriction. W e ﬁx an integer ℓ ≥ 2 ca lled level . Let t a be the num b er in (2.1). The level ℓ r estricted T-system for g (with the unit b oundary condition) is relations (2.4)–(2.10) naturally r estricted to { T ( a ) m ( u ) | a ∈ I , 1 ≤ m ≤ t a ℓ − 1 , u ∈ U } by imposing T ( a ) t a ℓ ( u ) = 1 . The level ℓ restric ted Y-s y stem for g is relatio ns (2.11)–(2.15) natura lly restricted to { Y ( a ) m ( u ) | a ∈ I , 1 ≤ m ≤ t a ℓ − 1 , u ∈ U } b y imp osing Y ( a ) t a ℓ ( u ) − 1 = 0. Note that for g nonsimply la ced, the ab ov e restr iction makes se nse a lso a t ℓ = 1 . The res ulting T and Y-sys tems b ecome equiv alent to the level t res tr icted T and Y-systems for A n with n = ♯ { a ∈ I | t a = t } under the rescaling of the spectral parameter u → u/ t . One ca n a lso c o nsider the level 0 case formally . See around (16.2). Example 2 .2. W e write down the level 2 r estricted T and Y-systems for A 2 : T (1) 1 ( u − 1) T (1) 1 ( u + 1) = 1 + T (2) 1 ( u ) , T (2) 1 ( u − 1) T (2) 1 ( u + 1) = 1 + T (1) 1 ( u ) , Y (1) 1 ( u − 1) Y (1) 1 ( u + 1) = 1 + Y (2) 1 ( u ) , Y (2) 1 ( u − 1) Y (2) 1 ( u + 1) = 1 + Y (1) 1 ( u ) . Thu s they are iden tica l. Example 2 .3. W e write down the level 2 r estricted T-system for C 2 : T (1) 1 ( u − 1 2 ) T (1) 1 ( u + 1 2 ) = T (1) 2 ( u ) + T (2) 1 ( u ) , T (1) 2 ( u − 1 2 ) T (1) 2 ( u + 1 2 ) = T (1) 1 ( u ) T (1) 3 ( u ) + T (2) 1 ( u − 1 2 ) T (2) 1 ( u + 1 2 ) , T (1) 3 ( u − 1 2 ) T (1) 3 ( u + 1 2 ) = T (1) 2 ( u ) + T (2) 1 ( u ) , T (2) 1 ( u − 1) T (2) 1 ( u + 1) = 1 + T (1) 2 ( u ) . Example 2 .4. Level ℓ res tricted T-system for A r − 1 has the form T ( a ) m ( u − 1) T ( a ) m ( u + 1) = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) + T ( a − 1) m ( u ) T ( a +1) m ( u ) for 1 ≤ a ≤ r − 1 and 1 ≤ m ≤ ℓ − 1 . It is inv ar iant under the sim ulta neous transformatio n T ( a ) m ( u ) 7→ T ( m ) a ( ± u + const) and r ↔ ℓ . The similar prop erty holds also for the lev el ℓ r e s tricted Y-system for A r − 1 . This symmetry is ca lled the level-r ank duality . 2.3. Relation b etw een T and Y-systems . The unres tricted T-system for g has the form T ( a ) m ( u − 1 t a ) T ( a ) m ( u + 1 t a ) = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) + Y ( b,k,v ) T ( b ) k ( v ) N ( a, m,u | b,k,v ) , (2.16) where the last term is a ﬁnite pro duct. Then, it is easy to see that the unrestricted Y-system for the same g takes the for m Y ( a ) m ( u − 1 t a ) Y ( a ) m ( u + 1 t a ) = Q ( b,k,v ) (1 + Y ( b ) k ( v )) N ( b, k ,v | a,m,u ) (1 + Y ( a ) m − 1 ( u ) − 1 )(1 + Y ( a ) m +1 ( u ) − 1 ) . (2.17) The same relation holds also b etw een the level ℓ restricted T and Y-systems. Let us write (2.16) simply as T ( a ) m ( u − 1 t a ) T ( a ) m ( u + 1 t a ) = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) + M ( a ) m ( u ) . (2.18 ) 14 Theorem 2. 5 ([1]) . Supp ose T ( a ) m ( u ) satisﬁes the unr est ricte d T-system for g . Then Y ( a ) m ( u ) = M ( a ) m ( u ) T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) (2.19) is a solution of the un r estricte d Y-system for g . The same claim holds b etwe en the level ℓ restricte d T and Y-syst ems. Sketch of pr o of . This can b e dir ectly veriﬁed by substituting the resulting r elations 1 + Y ( a ) m ( u ) = T ( a ) m  u − 1 t a  T ( a ) m  u + 1 t a  T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) , (2.20) 1 + Y ( a ) m ( u ) − 1 = T ( a ) m  u − 1 t a  T ( a ) m  u + 1 t a  M ( a ) m ( u ) (2.21) int o the Y-system. Here we de mo nstrate the calculation for simply laced g . Y ( a ) m ( u − 1) Y ( a ) m ( u + 1) = Q b : C ab = − 1 T ( b ) m ( u − 1) T ( b ) m ( u + 1) T ( a ) m − 1 ( u − 1) T ( a ) m +1 ( u − 1) T ( a ) m − 1 ( u + 1) T ( a ) m +1 ( u + 1) = Q b : C ab = − 1 ( T ( b ) m − 1 ( u ) T ( b ) m +1 ( u ) + Q c : C bc = − 1 T ( c ) m ( u )) T ( a ) m − 2 ( u ) T ( a ) m ( u ) + Q b : C ab = − 1 T ( b ) m − 1 ( u ) × 1 T ( a ) m ( u ) T ( a ) m +2 ( u ) + Q b : C ab = − 1 T ( b ) m +1 ( u ) = Q b : C ab = − 1 (1 + Y ( b ) m ( u )) (1 + Y ( a ) m − 1 ( u ) − 1 )(1 + Y ( a ) m +1 ( u ) − 1 ) . This calculation is v alid also at m = 1 by forma lly setting T ( a ) − 1 ( u ) = 0. F or level ℓ restricted case, it is v alid similarly by for mally setting T ( a ) ℓ +1 ( u ) = 0 . Theorem 2.5 has a natural ac c o unt from the viewpoint of cluster a lgebra with co eﬃcients. See Remark 5 .5. Example 2. 6. W e write down the rela tio n (2.19) for the level 2 restricted T- system for C 2 . F ro m Example 2.3, they read Y (1) 1 ( u ) = T (2) 1 ( u ) T (1) 2 ( u ) , Y (1) 2 ( u ) = T (2) 1 ( u − 1 2 ) T (2) 1 ( u + 1 2 ) T (1) 1 ( u ) T (1) 3 ( u ) , Y (1) 3 ( u ) = T (2) 1 ( u ) T (1) 2 ( u ) , Y (2) 1 ( u ) = T (1) 2 ( u ) . Thu s the sp eciﬁc constr uction (2.19) automatica lly impose s the condition Y (1) 1 ( u ) = Y (1) 3 ( u ). How ever, the level res tricted Y-sy s tem alone do e s not restrict itself to such a situation in general. Remark 2.7. Consider a slight mo diﬁcation of the ge neral T-system relation (2.18) int o T ( a ) m ( u − 1 t a ) T ( a ) m ( u + 1 t a ) = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) + g ( a ) m ( u ) M ( a ) m ( u ) , (2.22) 15 where g ( a ) m ( u ) is any function satisfying g ( a ) m ( u − 1 t a ) g ( a ) m ( u + 1 t a ) = g ( a ) m − 1 ( u ) g ( a ) m +1 ( u ) . (2.23) Then it is easily chec ked that the substitution Y ( a ) m ( u ) = g ( a ) m ( u ) M ( a ) m ( u ) T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) (2.24) is still a solution of the same Y-system. 2.4. Twisted case. Let us proceed to the T and Y-systems asso cia ted with the t wisted quantum aﬃne alge br as following [12, 13]. In this subsection and the next, X N exclusively denotes a Dynkin diag ram of type A N ( N ≥ 2 ), D N ( N ≥ 4 ) or E 6 . W e keep the enumeration of the no des of X N by the set I = { 1 , . . . , N } as in Figure 1. F o r a pair ( X N , κ ) = ( A N , 2), ( D N , 2), ( E 6 , 2) o r ( D 4 , 3), we deﬁne the diagram automorphism σ : I → I of X N of order κ a s follows: σ ( a ) = a exc ept for the fol lowing c ases in ou r enumera t ion : σ ( a ) = N + 1 − a ( a ∈ I ) ( X N , κ ) = ( A N , 2) , (2.25) σ ( N − 1) = N , σ ( N ) = N − 1 ( X N , κ ) = ( D N , 2) , σ (1) = 6 , σ (2) = 5 , σ (5) = 2 , σ (6) = 1 ( X N , κ ) = ( E 6 , 2) , σ (1) = 3 , σ (3) = 4 , σ (4) = 1 ( X N , κ ) = ( D 4 , 3) . Let I /σ b e the set of the σ -o rbits of no des of X N . W e cho ose, at our discr e tio n, a complete set of representativ es I σ ⊂ I of I /σ as I σ =      { 1 , 2 , . . . , r } ( X N , κ ) = ( A 2 r − 1 , 2) , ( A 2 r , 2) , ( D r +1 , 2) , { 1 , 2 , 3 , 4 } ( X N , κ ) = ( E 6 , 2) , { 1 , 2 } ( X N , κ ) = ( D 4 , 3) . (2.26) A (2) 2 r − 1 1 0 2 r − 1 r A (2) 2 0 1 A (2) 2 r 0 1 r − 1 r D (2) r +1 0 1 r − 1 r E (2) 6 0 1 2 3 4 D (3) 4 0 1 2 Figure 2. The Dynkin diagr ams X ( κ ) N of twisted a ﬃne type and their en umerations b y I σ ∪ { 0 } . F or a ﬁlled node a , σ ( a ) = a (i.e., κ a = κ ) holds. Let X ( κ ) N = A (2) 2 r − 1 ( r ≥ 2) , A (2) 2 r ( r ≥ 1) , D (2) r +1 ( r ≥ 3) , E (2) 6 or D (3) 4 be a Dynkin diagram of twisted aﬃne t y pe [11]. W e enumerate the no des of X ( κ ) N with I σ ∪ { 0 } as in Figure 2, where I σ is the one for ( X N , κ ). By this, we hav e established the ident iﬁcation of the non-0 th no des o f the diagram X ( κ ) N with the no des of the 16 diagram X N belo nging to the set I σ . F or example, for E (2) 6 , the corr esp ondence is as follows: E (2) 6 0 1 2 3 4 E 6 1 2 3 5 6 4 The ﬁlled no des 3,4 in E (2) 6 corres p o nd to the ﬁxed no des by σ in E 6 . W e use this ident iﬁcation throughout. (The 0th no de of X ( κ ) N is irrelev ant in our setting her e .) W e deﬁne κ a ( a ∈ I σ ) as κ a = ( 1 σ ( a ) 6 = a, κ σ ( a ) = a. (2.27) Note that X (2) N = A (2) 2 r is the unique case in which κ a = 1 for any a ∈ I σ . By U q ( X ( κ ) N ) we mean the quantized universal env eloping alg e br a [ 14] of the twisted aﬃne Lie algebra of type X ( κ ) N [11]. Let us pro ceed to the unrestr icted T-systems . Cho ose ~ ∈ C \ 2 π √ − 1 Q arbitrar - ily . The unrestricted T-sys tem for U q ( X ( κ ) N ) is the fo llowing relations for commut- ing v ar iables { T ( a ) m ( u ) | a ∈ I σ , m ∈ Z ≥ 1 , u ∈ C κ a ~ } , where Ω = 2 π √ − 1 /κ ~ , a nd T (0) m ( u ) = T ( a ) 0 ( u ) = 1 if they o ccur on the RHS in the relatio ns : F or X ( κ ) N = A (2) 2 r − 1 , T ( a ) m ( u − 1) T ( a ) m ( u + 1) = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) (2.28) + T ( a − 1) m ( u ) T ( a +1) m ( u ) (1 ≤ a ≤ r − 1) , T ( r ) m ( u − 1) T ( r ) m ( u + 1) = T ( r ) m − 1 ( u ) T ( r ) m +1 ( u ) + T ( r − 1) m ( u ) T ( r − 1) m ( u + Ω) . F or X ( κ ) N = A (2) 2 r , T ( a ) m ( u − 1) T ( a ) m ( u + 1) = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) (2.29) + T ( a − 1) m ( u ) T ( a +1) m ( u ) (1 ≤ a ≤ r − 1) , T ( r ) m ( u − 1) T ( r ) m ( u + 1) = T ( r ) m − 1 ( u ) T ( r ) m +1 ( u ) + T ( r − 1) m ( u ) T ( r ) m ( u + Ω) . F or X ( κ ) N = D (2) r +1 , T ( a ) m ( u − 1) T ( a ) m ( u + 1) = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) (2.30) + T ( a − 1) m ( u ) T ( a +1) m ( u ) (1 ≤ a ≤ r − 2) , T ( r − 1) m ( u − 1) T ( r − 1) m ( u + 1) = T ( r − 1) m − 1 ( u ) T ( r − 1) m +1 ( u ) + T ( r − 2) m ( u ) T ( r ) m ( u ) T ( r ) m ( u + Ω) , T ( r ) m ( u − 1) T ( r ) m ( u + 1) = T ( r ) m − 1 ( u ) T ( r ) m +1 ( u ) + T ( r − 1) m ( u ) . 17 F or X ( κ ) N = E (2) 6 , T (1) m ( u − 1) T (1) m ( u + 1) = T (1) m − 1 ( u ) T (1) m +1 ( u ) + T (2) m ( u ) , (2.31) T (2) m ( u − 1) T (2) m ( u + 1) = T (2) m − 1 ( u ) T (2) m +1 ( u ) + T (1) m ( u ) T (3) m ( u ) , T (3) m ( u − 1) T (3) m ( u + 1) = T (3) m − 1 ( u ) T (3) m +1 ( u ) + T (2) m ( u ) T (2) m ( u + Ω) T (4) m ( u ) , T (4) m ( u − 1) T (4) m ( u + 1) = T (4) m − 1 ( u ) T (4) m +1 ( u ) + T (3) m ( u ) . F or X ( κ ) N = D (3) 4 , T (1) m ( u − 1) T (1) m ( u + 1) = T (1) m − 1 ( u ) T (1) m +1 ( u ) + T (2) m ( u ) , (2.32) T (2) m ( u − 1) T (2) m ( u + 1) = T (2) m − 1 ( u ) T (2) m +1 ( u ) + T (1) m ( u ) T (1) m ( u − Ω) T (1) m ( u + Ω) . The domain C κ a ~ of the parameter u eﬀectively imp oses the following p erio dicity: T ( a ) m ( u ) = ( T ( a ) m ( u + κ Ω) σ ( a ) 6 = a, T ( a ) m ( u + Ω) σ ( a ) = a. (2.33) Remark 2 .8. The T-s y stem for U q ( X ( κ ) N ) is obta inable from the T-system for g = X N by a folding in the following sense. Denoting the v ariable in the latter by ˜ T ( a ) m ( u ) with a ∈ I , one imp oses the condition ˜ T ( σ k ( a )) m ( u ) = ˜ T ( a ) m ( u + k Ω) a nd ident iﬁes ˜ T ( a ) m ( u ) with a ∈ I σ ⊂ I as the v ariable T ( a ) m ( u ) in the former. The sa me remark applies also to the Y-system given in what follows. The unrestricted Y-system for U q ( X ( κ ) N ) is the follo wing relations for the com- m uting v ariables { Y ( a ) m ( u ) | a ∈ I σ , m ∈ Z ≥ 1 , u ∈ C κ a ~ } , where Ω = 2 π √ − 1 /κ ~ , and Y (0) m ( u ) = Y ( a ) 0 ( u ) − 1 = 0 if they o ccur on the RHS in the r elations: F or X ( κ ) N = A (2) 2 r − 1 , Y ( a ) m ( u − 1) Y ( a ) m ( u + 1) = (1 + Y ( a − 1) m ( u ))(1 + Y ( a +1) m ( u )) (1 + Y ( a ) m − 1 ( u ) − 1 )(1 + Y ( a ) m +1 ( u ) − 1 ) (2.34) (1 ≤ a ≤ r − 1) , Y ( r ) m ( u − 1) Y ( r ) m ( u + 1) = (1 + Y ( r − 1) m ( u ))(1 + Y ( r − 1) m ( u + Ω)) (1 + Y ( r ) m − 1 ( u ) − 1 )(1 + Y ( r ) m +1 ( u ) − 1 ) . F or X ( κ ) N = A (2) 2 r , Y ( a ) m ( u − 1) Y ( a ) m ( u + 1) = (1 + Y ( a − 1) m ( u ))(1 + Y ( a +1) m ( u )) (1 + Y ( a ) m − 1 ( u ) − 1 )(1 + Y ( a ) m +1 ( u ) − 1 ) (2.35) (1 ≤ a ≤ r − 1) , Y ( r ) m ( u − 1) Y ( r ) m ( u + 1) = (1 + Y ( r − 1) m ( u ))(1 + Y ( r ) m ( u + Ω)) (1 + Y ( r ) m − 1 ( u ) − 1 )(1 + Y ( r ) m +1 ( u ) − 1 ) . 18 F or X ( κ ) N = D (2) r +1 , Y ( a ) m ( u − 1) Y ( a ) m ( u + 1) = (1 + Y ( a − 1) m ( u ))(1 + Y ( a +1) m ( u )) (1 + Y ( a ) m − 1 ( u ) − 1 )(1 + Y ( a ) m +1 ( u ) − 1 ) (2.36) (1 ≤ a ≤ r − 2) , Y ( r − 1) m ( u − 1) Y ( r − 1) m ( u + 1) = (1 + Y ( r − 2) m ( u ))(1 + Y ( r ) m ( u ))(1 + Y ( r ) m ( u + Ω)) (1 + Y ( r − 1) m − 1 ( u ) − 1 )(1 + Y ( r − 1) m +1 ( u ) − 1 ) , Y ( r ) m ( u − 1) Y ( r ) m ( u + 1) = 1 + Y ( r − 1) m ( u ) (1 + Y ( r ) m − 1 ( u ) − 1 )(1 + Y ( r ) m +1 ( u ) − 1 ) . F or X ( κ ) N = E (2) 6 , Y (1) m ( u − 1) Y (1) m ( u + 1) = 1 + Y (2) m ( u ) (1 + Y (1) m − 1 ( u ) − 1 )(1 + Y (1) m +1 ( u ) − 1 ) , (2.37) Y (2) m ( u − 1) Y (2) m ( u + 1) = (1 + Y (1) m ( u ))(1 + Y (3) m ( u )) (1 + Y (2) m − 1 ( u ) − 1 )(1 + Y (2) m +1 ( u ) − 1 ) , Y (3) m ( u − 1) Y (3) m ( u + 1) = (1 + Y (2) m ( u ))(1 + Y (2) m ( u + Ω))(1 + Y (4) m ( u )) (1 + Y (3) m − 1 ( u ) − 1 )(1 + Y (3) m +1 ( u ) − 1 ) , Y (4) m ( u − 1) Y (4) m ( u + 1) = 1 + Y (3) m ( u ) (1 + Y (4) m − 1 ( u ) − 1 )(1 + Y (4) m +1 ( u ) − 1 ) . F or X ( κ ) N = D (3) 4 , Y (1) m ( u − 1) Y (1) m ( u + 1) = 1 + Y (2) m ( u ) (1 + Y (1) m − 1 ( u ) − 1 )(1 + Y (1) m +1 ( u ) − 1 ) , (2 .38) Y (2) m ( u − 1) Y (2) m ( u + 1) = (1 + Y (1) m ( u ))(1 + Y (1) m ( u − Ω))(1 + Y (1) m ( u + Ω)) (1 + Y (2) m − 1 ( u ) − 1 )(1 + Y (2) m +1 ( u ) − 1 ) . 2.5. Restriction and relations b etw een T and Y-systems. Fix an integer ℓ ≥ 2 called level . The le vel ℓ restric ted T-system fo r U q ( X ( κ ) N ) (with the unit bo undary condition) is the re la tions (2.28)–(2.32) natura lly restr ic ted to { T ( a ) m ( u ) | a ∈ I σ , 1 ≤ m ≤ ℓ − 1 , u ∈ C κ a ~ } b y impo sing T ( a ) ℓ ( u ) = 1 (the unit boundar y condition). The level ℓ restr icted Y-sys tem for U q ( X ( κ ) N ) is the r elations (2.34)–(2 .3 8) na t- urally restricted to { Y ( a ) m ( u ) | a ∈ I σ , 1 ≤ m ≤ ℓ − 1 , u ∈ C κ a ~ } b y imp osing Y ( a ) ℓ ( u ) − 1 = 1. The pro pe rties stated in Theore m 2.5 and Remark 2.7 also hold b etw een the T and Y-systems of for U q ( X ( κ ) N ). O n the other hand, the cor resp ondence like (2.16) and (2.17) in the unt wisted case is not v alid. 2.6. U q ( sl ( r | s )) case. Among a v ariety of Lie supe r a lgebras, we present the T- system and the Y-system r elated to U q ( sl ( r | s )) as a t ypical example. F o r brevity we employ the following notation within this subsection. H r,s = ( Z > 0 × Z > 0 ) \ ( Z ≥ r × Z ≥ s ) , H r,s = ( Z ≥ 0 × Z ≥ 0 ) \ ( Z >r × Z >s ) . (2.39) 19 These sets a re often called fat ho o k . The T-system fo r U q ( sl ( r | s )) is the following relations among the commut ing v ariables { T ( a ) m ( u ) | ( a, m ) ∈ H r,s , u ∈ U } . T ( a ) m ( u − 1) T ( a ) m ( u + 1) = T ( a − 1) m ( u ) T ( a +1) m ( u ) + T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) , (2.40) T ( r ) s +1 ( u ) = T ( r +1) s ( u ) . (2.41) Relation (2.40) is impo sed for all ( a, m ) ∈ H r,s \ { (0 , 0 ) } , w he r e if a ny T ( b ) k ( u ) with ( b, k ) 6∈ H r,s is contained in the RHS, it should be understo o d as 0. T ( b ) k ( u ) = 0 if ( b, k ) 6∈ H r,s . (2.42) This leads to the simple recursion relations for the sequences corresp onding to the bo undary H r,s \ H r,s . T ( a ) m ( u − 1) T ( a ) m ( u + 1) = T ( a ) m +1 ( u ) T ( a ) m − 1 ( u ) ( a, m ) ∈ ( r, Z >s ) ∪ (0 , Z > 0 ) , T ( a ) m ( u − 1) T ( a ) m ( u + 1) = T ( a − 1) m ( u ) T ( a +1) m ( u ) ( a, m ) ∈ ( Z >r , s ) ∪ ( Z > 0 , 0) . (2.43) The extra relation (2.41) leads by induction to T ( r ) s + a ( u ) = T ( r + a ) s ( u ) a ≥ 0 . (2.44) In the applications, the v aria bles a ppe a ring in (2 .4 3) and (2.44) a re chosen appr o- priately r e ﬂe c ting the normaliz ation of the s y stem. The r e lation (2 .40) is the same as t yp e A case. The essential diﬀerence from it lies in (2.42) and (2.44). Let us pro ceed to the Y-system. W e as sume r ≥ s ≥ 2 ﬁrst. The Y-system for U q ( sl ( r | s )) is the following relatio ns among the commut ing v aria bles { Υ ( a ) 1 ( u ) , Υ ( a ) 2 ( u ) | a ∈ Z ≥ 1 , u ∈ U } ∪ { Y ( a ) m ( u ) | ( a, m ) ∈ H r,s , u ∈ U } . Y ( a ) m ( u − 1) Y ( a ) m ( u + 1) = (1 + Y ( a ) m +1 ( u ))(1 + Y ( a ) m − 1 ( u )) (1 + Y ( a − 1) m ( u ) − 1 )(1 + Y ( a +1) m ( u ) − 1 ) ( a, m ) ∈ H r,s , (2.45) Υ (1) 1 ( u − 1)Υ (1) 1 ( u + 1) = Υ (2) 2 ( u )(1 + Y (1) s − 1 ( u )) , (2.46) Υ ( a ) 1 ( u − 1)Υ ( a ) 1 ( u + 1) = Υ ( a +1) 1 ( u )Υ ( a − 1) 1 ( u ) 1 + Y ( a ) s − 1 ( u ) 1 + Y ( a − 1) s ( u ) a ≥ 2 , (2.47) Υ ( a ) 2 ( u − 1)Υ ( a ) 2 ( u + 1) = Υ ( a +1) 2 ( u )Υ ( a − 1) 2 ( u )(1 + Y ( a ) s − 1 ( u )) a ≥ 2 , (2.48) Υ (1) 1 ( u ) = Υ (1) 2 ( u ) , Υ ( r ) 1 ( u ) = Y ( r ) s ( u ) . (2.49) On the RHS of these relatio ns, any factor (1 + Y ( b ) k ( u ) ± 1 ) with ( b , k ) 6∈ H r,s is to be understo o d as 1. When r > s = 1, the equa tions (2 .46) and (2.48) a r e a bsent. The Y-system for s ≥ r ≥ 2 is given by (2 .45)–(2.49) by interc hanging r and s . 20 There is a simple relatio n b etw een the T-system and Y-system analo gous to Theorem 2.5. Supp ose that T ( a ) m ( u ) is a so lution to the T-sys tem. Then the com- binations Y ( a ) m ( u ) = T ( a ) m +1 ( u ) T ( a ) m − 1 ( u ) T ( a +1) m ( u ) T ( a − 1) m ( u ) ( a, m ) ∈ H r,s , (2.50) Υ ( a ) 1 ( u ) = T ( a ) s − 1 ( u ) T ( a − 1) s ( u ) , Υ ( a ) 2 ( u ) = T ( a ) s − 1 ( u ) T (0) s + a − 1 ( u ) (2.51) satisfy the Y-sy stem. In particula r, (2.4 9) ho lds due to (2.41). When s ≥ r ≥ 2, the parallel fact holds by in terchanging r and s and the r ole of indices a a nd m in T ( a ) m ( u ) and Y ( a ) m ( u ) ev erywhere. In view of the symmetry o f the sets (2.3 9), we do not in tr o duce the level r e striction. Remark 2.9. The ab ov e s et of relations seems diﬀerent fro m those given in [1 5] for g l (2 | 2 ), where a sp ecia l rela tio n T ( r ) s − 2 ∝ T ( r − 2) s v alid only for this case is uti- lized. Tha nks to this, Υ ( a ) 1 ( a 6 = r ) and Υ ( a ) 2 are not necessar ily needed. The tw o sets of Y-systems nevertheless lea d to an ident ical set o f ther mo dynamic Bethe ansatz equations 6 . The Y-system (2 .45)–(2.49) is consisten t with the thermody- namic Bethe ansatz equations in [16] under the identiﬁcation N , K ↔ r , s a nd Y ( a ) s − m = e − ζ ( a ) m /T (1 ≤ a, 1 ≤ m ≤ s − 1) , Y ( a ) s = e − ǫ a /T (1 ≤ a ≤ r ) , Y ( a ) s + j = e κ ( j ) a /T (1 ≤ j, 1 ≤ a ≤ r − 1) . 2.7. Bibliog raphical notes. The Hiro ta relation (2.5) for transfer matrices in the A r case ﬁrst a pp e ared in [1], where the T-sys tem for g was introduce d as functional relations a mong the commuting transfer matr ic es { T ( a ) m ( u ) } . The mo dels rele v a nt to the unres tricted and restr icted versions ar e the vertex and the restricted so lid- on-solid (RSOS) type mo dels, re s pe c tively . In such a setting, T-sy s tem a c quires some sca lar co eﬃcients dep ending on the normaliza tion of T ( a ) m ( u ) as in Remark 2.7. The unit bounda ry condition is also mo diﬁed according ly . Actually in [1], the restric ted T-system was in tro duced b y imp osing a slig ht ly weaker condition T ( a ) t a ℓ +1 ( u ) = 0. The T-system for the twisted case was in tr o duced in [1 2] in a similar context. Our presentation here follows [1 7, 1 3]. The T-system uniﬁes the many functional relations studied earlier individually . See Sections 3-4 for mor e details. The level ℓ restricted Y-system for g was intro duce d in [3] for simply laced g with ℓ = 2 as a universal prope r ty o f the thermo dynamic Bethe ansatz (TBA) equation in the cont ext of in tegr able p er turbations of conformal ﬁeld theories. Then, it was extended to the gener a l case in [4] based on the TBA equation re la ted to RSO S mo dels for U q ( ˆ g ) [1 8]. This pro cedure is detailed in Section 14. The Y-system for s imply lace d g was also given in [5] indepe ndently . F or more litera tures in the similar con tex t, see Section 14.7. The transformation (2.19) betw e en the T and Y-systems ﬁrst app eared in [7] for the simplest c a se g = A 1 , and ex tended in [1] to general g . T-systems related to Lie super alg e bras and super symmetric mo dels hav e b een studied in v arious contexts. See for example [15, 1 9, 20 , 21, 22, 23, 2 4] and references therein. 6 There are typos in [15] for g l (2 | 2), around (5.4) and (5.5). 21 3. T-system among commuting transfer ma trices The a im of this sectio n is to intro duce the basic e x amples of solv able lattice mo dels, b oth vertex a nd restricted solid-on-solid (RSOS) t yp e, and demonstrate how the T-system is obta ine d for their tra nsfer matrices in co nnection to the fusion pro cedure. Although thes e issues a re no wadays well recognized to b e in timately related to the representation theory of qua nt um groups , we defer s uch a descr iption to Section 4 avoiding to o many deﬁnitions fr om the beginning. O ur presentation here is based on explicit calculatio ns in trigo nometric par a meterization a long the simplest example fro m g = A 1 The exceptio n is the las t s ubsection 3 .7, where we will formally a rgue the general features of those mo dels asso cia ted with g eneral g quoting known facts on Kirillov-Reshetikhin mo dules and Q-system from Sections 4, 13.6 and 14.6. 3.1. V ertex mo dels and fusion. W e reca ll the 6 vertex mo del a nd its fusion without m uch reco urse to the repre s entation theor y 7 . Consider the tw o dimensional square lattice, where ea ch edge is assigned with a lo ca l v ariable b elonging to { 1 , 2 } . Around each vertex, we allo w the following 6 conﬁgur ations with the resp ective Boltzmann weigh ts. 1 1 1 1 2 2 2 2 1 2 2 1 2 1 1 2 2 1 2 1 1 2 1 2 1 − q 2 z 1 − q 2 z q (1 − z ) q (1 − z ) z (1 − q 2 ) 1 − q 2 . (3.1) The o ther 10 conﬁgura tions ar e a ssigned with 0 Boltzmann weigh t. Let V = C v 1 ⊕ C v 2 . Then (3.1) is a rrange d in the quantum R matr ix R ( z ) ∈ End( V ⊗ V ) as R ( z ) = a ( z ) X i E ii ⊗ E ii + b ( z ) X i 6 = j E ii ⊗ E j j + c ( z )  z X ij  E j i ⊗ E ij , a ( z ) = 1 − q 2 z , b ( z ) = q (1 − z ) , c ( z ) = 1 − q 2 . (3.2) Here the indices run over { 1 , 2 } a nd E ij is the matr ix unit acting as E ij v k = δ j k v i . The R matrix R ( z ) is a sso ciated with the quantu m a ﬃne a lgebra U q = U q ( A (1) 1 ) [14]. In fact, ˇ R ( z ) := P R ( z ) commutes with ∆( U q ), where P denotes the transp osition of the compone nts 8 . A more deta iled account will b e given in Sec tion 4.3. Schematically (3.2 ) is expressed as R ( z ) = X ij k l  l j z i k  E ij ⊗ E kl , ˇ R ( z ) = X ij k l  l j z k i  E ij ⊗ E kl , (3.3) where the z dep endence is exhibited. The Y ang -Baxter equation R 23 ( z ′ ) R 13 ( z ) R 12 ( z / z ′ ) = R 12 ( z / z ′ ) R 13 ( z ) R 23 ( z ′ ) 7 Some terminology will b e reﬁned after (3.16). 8 The asymmetry b et we en the last tw o in (3.1 ) is due to our choice of the copro duct (4.9). It ﬁts the crystal base theory making the limit q → 0 of (3.7) w ell deﬁned, although this fact will not b e used in this review. 22 holds [2 ], where the indices signify the comp onents in the tenso r pr o duct a s 1 V ⊗ 2 V ⊗ 3 V on which the b oth s ides act. It is depicted a s z / z ′ z z ′ = z ′ z / z ′ z (3.4) Starting from the 6 vertex mo del [2 5, 26], one ca n co nstruct higher spin solv able vertex mo dels by the fusion pro cedure [27]. Let V m be the ir reducible U q mo dule spanned by the m fold q − symmetric tenso r s. Concretely , V 1 = V and V m with m ≥ 2 is realized as the quotient V ⊗ m / A , w her e A = P j V ⊗ j ⊗ Im ˇ R ( q − 2 ) ⊗ V ⊗ m − 2 − j . It is ea sy to s e e Im ˇ R ( q − 2 ) = K er ˇ R ( q 2 ) = C ( v 1 ⊗ v 2 − q v 2 ⊗ v 1 ). W e take the base vector o f V m as v ⊗ x 2 2 ⊗ v ⊗ x 1 1 mo d A , where x i ∈ Z ≥ 0 and x 1 + x 2 = m . The base will also be denoted by x = ( x 1 , x 2 ) for brevity . Obviously dim V m = m + 1. The Y ang- Baxter e quation (3.4) with z ′ = z q 2 shows that Im ˇ R ( q − 2 ) ⊂ 1 V ⊗ 2 V is preserved under the a ction of R 13 ( z q 2 ) R 23 ( z ). Ther efore its action on ( 1 V ⊗ 2 V ) ⊗ 3 V can b e restricted to V 2 ⊗ V 1 =  ( V ⊗ V ) / Im ˇ R ( q − 2 )  ⊗ V . Similarly , by using (3 .4) rep eatedly , it is shown tha t the comp osition R 1 ,m +1 ( z q m − 1 ) R 2 ,m +1 ( z q m − 3 ) · · · R m,m +1 ( z q − m +1 ) a ( z q m − 3 ) a ( z q m − 5 ) · · · a ( z q − m +1 ) (3.5) can b e restricted to V m ⊗ V 1 . The resulting oper ator, the fusion R matrix R ( m, 1) ( z ) ∈ End( V m ⊗ V 1 ), is given by R ( m, 1) ( z )( x ⊗ v j ) = X k =1 , 2  j x z y k  y ⊗ v k , (3.6) j x z y k =      q m − x k − q x k +1 z j = k , (1 − q 2 x 1 ) z ( j, k ) = (2 , 1) , 1 − q 2 x 2 ( j, k ) = (1 , 2) , (3.7) where y = ( y 1 , y 2 ) is sp eciﬁed by the weigh t conserv ation (so ca lled “ice rule”) as y i = x i + δ ij − δ ik . By the deﬁnition R (1 , 1) ( z ) = R ( z ) and (3.7) reduces to (3.1) for m = 1 . In the case ( j, k ) = (1 , 2) for example, the matrix element 1 − q 2 x 2 is obtained from the following ca lculation ( D =denominato r in (3.5)): 1 D P x 2 i =1 q i − 1 z q − m +1 . . . . . . z q m − 1 x 1 i − 1 x 2 − i = x 2 X i =1 q i − 1 (1 − q 2 ) a ( z q m − 1 ) Q x 1 + i − 1 n = x 1 +1 b ( z q − m − 1+2 n ) Q x 1 + i n = x 1 +1 a ( z q − m − 1+2 n ) . (3.8) 23 The red and blue edg es are ass ig ned with the lo cal states 1 and 2 , r esp ectively . The incoming state (left column) r epresents v ⊗ x 2 2 ⊗ v ⊗ x 1 1 . The facto r q i − 1 accounts for the eﬀect o f rearr anging the outgoing state into the base form by using the rela tion v 1 ⊗ v 2 ≡ q v 2 ⊗ v 1 mo d A as v ⊗ x 2 − i 2 ⊗ v 1 ⊗ v ⊗ i − 1 2 ⊗ v ⊗ x 1 1 ≡ q i − 1 v ⊗ y 2 2 ⊗ v ⊗ y 1 1 ∈ V m , where y = ( y 1 , y 2 ) = ( x 1 + 1 , x 2 − 1) for ( j, k ) = (1 , 2). One can fuse R ( m, 1) ( z ) further along the other compo nent of the tensor pro duct in a completely parallel fashion. The comp osition R ( m, 1) 0 ,n ( z q n − 1 ) · · · R ( m, 1) 0 , 2 ( z q − n +3 ) R ( m, 1) 0 , 1 ( z q − n +1 ) ∈ E nd( V m ⊗ V ⊗ n 1 ) (3.9) can b e restricted to V m ⊗ V n . The re sult yields the quantum R matrix R ( m,n ) ( z ) ∈ End( V m ⊗ V n ). The R ma trices so obtained aga in satisfy the Y ang-Baxter equatio n in End( V l ⊗ V m ⊗ V n ): R ( m,n ) 23 ( z ′ ) R ( l,n ) 13 ( z ) R ( l,m ) 12 ( z / z ′ ) = R ( l,m ) 12 ( z / z ′ ) R ( l,n ) 13 ( z ) R ( m,n ) 23 ( z ′ ) . (3.10) It is depicted as (3 .4) with the thre e lines to b e interpreted as r epresenting V l , V m and V n . The q ua nt um R ma trix R ( m,n ) ( z ) gives rise to a fusion vertex mo del on a planar square la ttice by the same rule as diagrams (3.3) and (3.6). The loca l v ariables on the horizontal and vertical edg es are taken fro m V m and V n , resp ectively . 3.2. T ransfer matrices. Here we use the additiv e spectr al parameter u as well as the m ultiplicative one z . They are related as z = q u . W e introduce the row to row transfer matrix T m ( u ) = T r V m  R ( m,s N ) 0 ,N ( z / w N ) · · · R ( m,s 1 ) 0 , 1 ( z / w 1 )  = X x ∈ V m x x . z /w 1 · · · z /w N (3.11) The hor izontal line is asso ciated with V m which is called the a uxiliary s pace. The trace over it co rresp onds to the p erio dic b oundary c o ndition. There are N vertical lines corre s po nding to V s 1 ⊗ · · · ⊗ V s N which is called the quantum s pace. The T m ( u ) is a linear op erator acting on the quantum spa c e. The data s i , w i represent the inhomogeneity in the spins and co upling constants. The ﬁrst consequence of the Y ang-Ba xter equatio n (3.10) is the commutativit y of the tr a nsfer matrices a cting on the co mmon qua nt um space (common s i and w i in the present context) [ T m ( u ) , T n ( v )] = 0 . (3.12) Let us take s i = 1 for all i for simplicity and demonstrate the functional relation T 1 ( u + 1) T 1 ( u − 1) = T 0 ( u ) T 2 ( u ) + g 1 ( u )id , T 0 ( u ) = N Y i =1 a ( z i /q ) , g 1 ( u ) = N Y i =1 a ( z i q ) b ( z i /q ) , (3.13) where z i = z /w i . This corre sp o nds to the T- system for A 1 (2.6) with m = 1 mo diﬁed by a mo del dep endent factors T 0 ( u ) and g 1 ( u ). Consider the diagr am for 24 T 1 ( u + 1) T 1 ( u − 1) corresp onding to the matrix element for the transition v α 1 ⊗ · · · ⊗ v α N 7→ v β 1 ⊗ · · · ⊗ v β N : X k,l =1 , 2 k k l l α 1 α N z 1 / q · · · z N / q z 1 q · · · z N q β 1 β N (3.14) Given α i , β i , the sum o ver k , l is r egarded as the trace of an operato r acting on the auxiliary spa ce V 1 ⊗ V 1 horizontally . The space V 1 ⊗ V 1 po ssesses the in v ar iant subspace Im ˇ R ( q − 2 ) = C ( v 1 ⊗ v 2 − q v 2 ⊗ v 1 ) which pro pagates to the right owing to the Y ang -Baxter e quation (3 .4). In fact, the following iden tity c an b e chec ked directly . 2 1 1 2 α k l z / q z q β α k l z / q z q β − q × = δ αβ a ( z q ) b ( z /q ) ×      1 ( k , l ) = (2 , 1) , − q ( k, l ) = (1 , 2) , 0 otherwise . (3.15) Thu s Im ˇ R ( q − 2 ) con tributes to T r V 1 ⊗ V 1 (3.14) as Q N i =1 δ α i ,β i a ( z i q ) b ( z i /q ), g iving the second term in the RHS of (3.13). The other contribution to the trace is from ( V 1 ⊗ V 1 ) / Im ˇ R ( q − 2 ) = V 2 . This is equal to T 0 ( u ) T 2 ( u ) b y the deﬁnition, where the fa ctor T 0 ( u ) is due to the denominator in (3.5) with m = 2. In this way o ne observes that the exact sequence 0 → Im ˇ R ( q − 2 ) → V 1 ⊗ V 1 → V 2 → 0 (3.16) plays a key ro le in deriving (3.13). In Section 4.2, we will int ro duce the Kirillov-Reshetikhin mo dule W ( a ) m ( u ) for general q uantum aﬃne a lgebra U q ( ˆ g ). The case g = A 1 relev ant here, denoted by W m ( u ) = W (1) m ( u ), will b e describ ed explicitly in Section 4 .3. In such a formalis m, one endows each line in the diag rams like (3.14)–(3.15) with a sp ectral para meter z = q u which corresp o nds to a Kirillov-Reshetikhin mo dule W m ( u ). The R ma- trix R ( m,n ) ( z ) ∈ End( V m ⊗ V n ) is actually to b e understo o d as R ( m,n ) ( z 1 /z 2 ) ∈ End( W m ( u 1 ) ⊗ W n ( u 2 )) with z i = q u i . Up to an ov era ll scalar, it is characterized by the intert w ining prop erty ∆( g ) P R ( m,n ) ( z 1 /z 2 ) = P R ( m,n ) ( z 1 /z 2 )∆( g ) wher e g is any element from U q ( A (1) 1 ) and ∆ is the copro duct (4.9 ) [1 4]. Accor dingly , w e say that the transfer matr ix T m ( u ) (3.1 1) has the a uxiliary space W m ( u ) and acts on the quantum space W s 1 ( v 1 ) ⊗ · · · ⊗ W s N ( v N ) with w i = q v i . The exact sequence (3.1 6) will a lso be reﬁned into the one among tensor pr o d- uct of K irillov-Reshetikhin mo dules. See (4.16). The T-system relation T m ( u + 1) T m ( u − 1) = T m +1 ( u ) T m − 1 ( u ) + g m ( u )id for g e ne r al m follows from Theorem 4 .2 with n = j = m . An additional feature her e is that one actually needs to co nsider the cen tr a l extens ion of U q ( A (1) 1 ) to prop erly cop e with the fac to r g m ( u ). W e refer to [1, section 2.2] for this p o int. See also [28]. T o summarize, the Kirillov-Reshetikhin mo dule of the quantum aﬃne algebra and their exa ct seq ue nc e for m the represe ntation theoretica l background fo r the R matrix, fusion proc edure and the T-system among comm uting family of transfer matrices. 25 3.3. Restricted soli d-on-sol i d (RSOS) mo dels and fusio n. Be sides vertex mo dels, there is a no ther cla ss o f s olv able la ttice mo dels called Interaction Round F ace (IRF or simply fac e ) mo dels [2 ]. The relatio n of the t wo classes of mo dels has bee n studied from v arious viewp oints [29, 30, 31, 32, 33]. Here we recall the 8 vertex solid-on-s o lid (8VSOS) mo del [34]. It is the fundamental exa mple as s o ciated with U q ( A (1) 1 ) at q a ro ot o f unity and s erves as the prototype of restr icted solid-o n-solid (RSOS) mo dels. It gener a lizes to U q ( ˆ g ) for an y g in principle. W e illustrate the fusion pro cedure [35] a nd the deriv ation of the simplest case of the T-s ystem for the commuting transfer matric e s [36, 37]. The conten ts are parallel with the 6 vertex mo del discuss e d in the previous subsection. F or simplicity we concentrate on the critical case 9 . Consider the tw o dimensional squar e lattice, wher e ea ch site is as signed with a lo cal state b elo nging to Z . On the tw o loca l states a, b on neig hboring sites, the condition | a − b | = 1 is imp osed. With the a llow ed conﬁguration round a face, the following Boltzmann weigh ts ar e as signed [34]. W  a a ∓ 1 a ± 1 a     u  = [2 + u ] q 1 / 2 [2] q 1 / 2 , W  a ± 1 a a a ± 1     u  = [2 ξ + 2 a ∓ u ] q 1 / 2 [2 ξ + 2 a ] q 1 / 2 , W  a ± 1 a a a ∓ 1     u  = [2 ξ + 2 a ± 2 ] q 1 / 2 [ u ] q 1 / 2 [2 ξ + 2 a ] q 1 / 2 [2] q 1 / 2 , (3.17) where u is the sp ectra l parameter , q and ξ a re ge neric co nstants which will be sp ecialized when considering the res triction in Section 3.5. The function [ u ] q 1 / 2 is given by r eplacing q → q 1 / 2 in [ u ] q = q u − q − u q − q − 1 . (3.18) The Boltzmann w e ights (3.17) ar e depicted as b c a d u = W  b c a d     u  . (3.19) It satisﬁes the (generalized) star -triangle relation [2] whic h plays the role of the Y ang-Ba xter equatio n in face mo dels: X g W  f g a b     u  W  e d f g     v  W  g d b c     u − v  = X g W  e d g c     u  W  g c a b     v  W  f e a g     u − v  . (3.20) 9 The R SOS mo dels allo w elli ptic Boltzmann weigh ts in general. The critical case means the trigonometric case of them. The fusi on pro cedure and the T-system are equally v al id in the elli ptic case as well. 26 The sum ov er g co nsists of at mo st tw o terms in each side b ecause o f the neighbor ing condition, e.g. | f − g | = | b − g | = | d − g | = 1 for the LHS. W e depict (3.20) as • v u − v u e d f c a b e d f c a b = • u u − v v (3.21) where • stands for the sum over the lo ca l state. The face s drawn tog ether a re to be understo o d as the pro duct of the attached Boltzmann w eig ht s. One ca n apply the fusion pro cedure to the 8VSOS mo del [35]. Note the pro per ties − 2 • = 0 ∴ a +1 − a − 1 a a • u − 1 u +1 a a • u − 1 u +1 ∝ a a − 2 • • a a u − 1 u +1 = a a u − 1 u +1 • • − 2 = 0 (3.22) where the seco nd equality from the right is due to the star -triangle rela tion. This implies that for m = 2 , the quantit y • • • c b α 1 α 2 . . . . . . α m − 1 a d u − m +1 u − m +3 u + m − 1 (3.23) is indep endent of α 1 , . . . , α m − 1 as long as they are chosen so tha t | α i − α i +1 | = 1 ( α 0 = b , α m = a ). The indep endence for g eneral m ca n b e shown similarly . Thus (3.23) only dep ends o n the lo cal states a, b, c, d o n the cor ne r s. W e deﬁne the fused Boltzmann weight W m, 1  b c a d     u  to be (3 .23) divide d by Q m − 1 j =1 [ u + m +1 − 2 j ] q 1 / 2 [2] q 1 / 2 . By induction on m , the following formulas are easily established ( W 1 , 1 = W ). W m, 1  b b ∓ 1 a a ∓ 1     u  = [2 ξ + a + b ± m ] q 1 / 2 [1 ± ( a − b ) + u ] q 1 / 2 [2] q 1 / 2 [2 ξ + 2 a ] q 1 / 2 , W m, 1  b b ± 1 a a ∓ 1     u  = [ m ± ( a − b )] q 1 / 2 [2 ξ + a + b ± 1 ± u ] q 1 / 2 [2] q 1 / 2 [2 ξ + 2 a ] q 1 / 2 . (3.24) One can fuse them further in the horizontal dir ection. A similar argument shows that the quant it y β 1 β 2 · · · β n − 1 b c a d · · · u − n +1 u − n +3 u + n − 1 • • • (3.25) 27 is indep endent of β 1 , . . . , β n − 1 as lo ng a s | β i − β i +1 | = 1 ( β 0 = b, β n = c ). Here ea ch rectangle stands for the weigh t W m, 1 (3.24) with the sp e c iﬁed sp ectral pa rameters. Thu s w e deﬁne W m,n  b c a d     u  to b e (3.25). By co ns truction, it is zero unless b − a, c − d ∈ {− m, − m + 2 , . . . , m } , c − b, d − a ∈ {− n, − n + 2 , . . . , n } . (3.2 6) The the star-triang le r elation (3.20) is generalized to X g W l,n  f g a b     u  W m,n  e d f g     v  W l,m  g d b c     u − v  = X g W l,n  e d g c     u  W m,n  g c a b     v  W l,m  f e a g     u − v  . (3.27) 3.4. Relation to vertex mo del s. The trigonometr ic fa c e mo dels under con- sideration a re r elated to the 6 vertex mo del and its fusion in Section 3.1 [30]. Let us explain it along the s implest cases (3.1 7) and (3.2). Let a ∈ Z ≥ 1 and V a − 1 be the spin a − 1 2 representation of U q ( A 1 ) in Section 3.1 10 . W e us e co- pro duct (4.9) and the concr ete form (4.10). In the irreducible decomp osition V a − 1 ⊗ V 1 = L b = a ± 1 V b − 1 , the hig hes t weigh t vector v a,b ∈ V b − 1 is given by v a,a +1 = v a − 1 1 ⊗ v 1 1 and v a,a − 1 = v a − 1 1 ⊗ v 1 2 − q a − 1 v a − 1 2 ⊗ v 1 1 . Repea ting this once more, one ge ts the highest weigh t vectors v a,b,c in the irreducible comp onent V c − 1 in the decomp ositio n of V a − 1 ⊗ V 1 ⊗ V 1 lab eled with a, b, c such that | a − b | = | b − c | = 1. Explicitly , they rea d v a,a +1 ,a +2 = v a − 1 1 ⊗ v 1 1 ⊗ v 1 1 , v a,a − 1 ,a = [ a − 1] q ( v a − 1 1 ⊗ v 1 2 ⊗ v 1 1 − q a − 1 v a − 1 2 ⊗ v 1 1 ⊗ v 1 1 ) , v a,a +1 ,a = [ a ] q v a − 1 1 ⊗ v 1 1 ⊗ v 1 2 − q a − 1 [ a − 1] q v a − 1 2 ⊗ v 1 1 ⊗ v 1 1 − q a v a − 1 1 ⊗ v 1 2 ⊗ v 1 1 , v a,a − 1 ,a − 2 = v a − 1 1 ⊗ v 1 2 ⊗ v 1 2 − q a − 1 v a − 1 2 ⊗ v 1 1 ⊗ v 1 2 − q a − 2 v a − 1 2 ⊗ v 1 2 ⊗ v 1 1 + q 2 a − 4 v a − 1 3 ⊗ v 1 1 ⊗ v 1 1 . (3.28) Now co nsider the op erator 1 ⊗ ˇ R ( z ) acting on V a − 1 ⊗ V 1 ⊗ V 1 . Since it co mm utes with U q ( A 1 ), the images of the highest weigh t vectors are again highest. The face Boltzmann weights c an b e ex tracted from the matrix elements b etw een those highest weigh t vectors as  1 ⊗ ˇ R ( q u )  v a,b,c = − ( q − q − 1 ) q 1+ u 2 X d W  b c a d     u  v a,d,c . (3.29 ) Here ξ = 0 in the RHS and the s um is ov er d such that | a − d | = | d − c | = 1. A similar relation holds also b etw een the fusion mo dels. Conv erse ly , one can deduce the R ma trix from the face Boltzmann weigh ts as a limit where the site v ariables or eﬀectiv ely ξ tends to inﬁnit y . F or instance, (3.7) is 10 Actually , V a − 1 can b e the V erma module with the highest we i gh t vect or v a − 1 1 suc h that k 1 v a − 1 1 = q a − 1 v a − 1 1 for generic a . 28 obtained from (3.24) as − ( q − q − 1 ) q m +1+ u 2 ( a, b ) m ( b, c ) 1 ( d, c ) m ( a, d ) 1 lim q ξ → 0 W m, 1  b c a d     u  = j x z y , k (3.30) ( a, b ) m = q 1 8 ( a − b ) 2 + 1 4 m ( a + b ) , z = q u , (3.31) x = ( x 1 , x 2 ) =  m − a + b 2 , m + a − b 2  , j = 3 + b − c 2 , k = 3 + a − d 2 . (3.32 ) The factor on the LHS of (3.30) do es not sp oil the star -triangle relation. 3.5. Restriction. The (fusion) face mo dels constructed thus far po s sess lo cal states ranging o ver the inﬁnite set Z and are ca lled unres tricted. T o obtain a mo del with ﬁnitely many lo cal s tates, we make r estriction . W e introduce the integer ℓ ∈ Z ≥ 2 called level , and sp ecialize the parameter s a s follows: ξ = 0 , q = exp  π √ − 1 ℓ + 2  , [ u ] q 1 / 2 = sin π u 2( ℓ +2) sin π 2( ℓ +2) . (3.33) W e further set W m,n  b c a d     u  = 0 unless the pair s ( a, b ) , ( d, c ) (resp. ( a, d ) , ( b, c )) are m -admissible (resp. n - admissible). W e say tha t a pair ( a, b ) is m - admissible if b − a ∈ {− m, − m + 2 , . . . , m } , (3.34) a + b ∈ { m + 2 , m + 4 , . . . , 2 ℓ + 2 − m } . (3.35) Notice that the admiss ibility forces a, b ∈ { 1 , 2 , . . . , ℓ +1 } . The res ulting W m,n  b c a d     u  with a, b, c, d ∈ { 1 , 2 , . . . , ℓ + 1 } is called the restricted B oltzmann weigh t. O ne may w o nder if [0] q 1 / 2 = [2 ℓ + 4] q 1 / 2 = 0 may cause a divergence somewher e in the c onstruction. How ever it has b een pro ved [35] that the r estricted Boltzmann weigh ts ar e well-deﬁned and satisfy the star-tr iangle rela tion (3.2 7) amo ng them- selves 11 . In this wa y one obtains the level ℓ RSOS mo del whose lo cal states be lo ng to { 1 , 2 , . . . , ℓ + 1 } a nd the fusion degree spec iﬁe d by m and n . Let us comment on the a dmissibility co ndition a mong which the ﬁrst o ne (3.3 4) already app eared in (3.26). When ℓ → ∞ , the admissibility reduces to the Clebsch- Gordan rule: V a − 1 ⊗ V m = M b − 1= | a − 1 − m | ,...,a + m − 3 ,a + m − 1 V b − 1 . (3.36) The RHS contains precisely those b such that ( a, b ) is m -admissible a t ℓ = ∞ . F or ℓ ﬁnite, the necessity of a + b ≤ 2 ℓ + 2 − m can b e seen for example in the ﬁrst Boltzmann weigh t in (3.24). Under the s p ecia lization (3.33), it con tains the factor sin  π ( a + b + m ) 2 ℓ +4  in the numerator. Thus the “next”’ b for whic h a + b = 2 ℓ + 4 − m “can no t b e reached”. Such a tr uncation is also observed at the level of characters ass o ciated with (3.36). Denoting the q -dimension of V a − 1 at ro ot o f unit y by dim q V a − 1 = sin  π a ℓ +2  / s in  π ℓ +2  , w e hav e (dim q V m )(dim q V a − 1 ) = X b :( a,b ) is m -admissible dim q V b − 1 . (3.37) 11 Actually the statemen t holds for appropriately symmetrized W m,n . See [35, section 2.2]. 29 This truncated decomp osition is a lso kno w n as the fusion rule in the SU(2) level ℓ WZW conformal ﬁeld theory [38]. Finally we remark that given ℓ , one can not fuse to o muc h. In fact, (3 .34) and (3.35) ﬁx the a dmissible pairs to { ( a, a ) | 1 ≤ a ≤ ℓ + 1 } at m = 0 and to { ( a, ℓ + 2 − a ) | 1 ≤ a ≤ ℓ + 1 } at m = ℓ . They lea d to completely frozen mo dels. Nontrivial situa tio ns corr esp ond to the fusion degrees in the range 1 ≤ m ≤ ℓ − 1. This is an orig in o f the truncation condition in the restricted T-system (Section 2.2) for g = A 1 . 3.6. T ransfer matrices. W e co nsider the row to row transfer matrix T m ( u ) with per io dic bo undary condition whose elements T m ( u ) b 1 ,...,b N a 1 ,...,a N are given by W m,s 1  b 1 b 2 a 1 a 2     u − v 1  · · · W m,s N − 1  b N − 1 b N a N − 1 a N     u − v N − 1  W m,s N  b N b 1 a N a 1     u − v N  . No sum is in volved. It is depicted a s T m ( u ) b 1 ,...,b N a 1 ,...,a N = b 1 b 2 b N − 1 b N b 1 u − v 1 · · · u − v N − 1 u − v N a 1 a 2 a N − 1 a N a 1 (3.38) Here ( a i , a i +1 ) , ( b i , b i +1 ) are s i -admissible ( a N +1 = a 1 , b N +1 = b 1 ) and ( a i , b i ) is m -admissible for all i . The inhomogeneity s i , v i in fusio n deg rees and co upling constants ar e ﬁxed and s uppressed in the notation. The T m ( u ) is zero unless the parity condition P N i =1 s i ≡ 0 mod 2 is satisﬁed. The star-tria ngle rela tion (3.27) implies the commut ativity [2] [ T m ( u ) , T n ( v )] = 0 . (3.39) Let us take s 1 = 1 for a ll i for simplicity and demo nstrate the functional r elation T 1 ( u + 1) T 1 ( u − 1) = T 0 ( u ) T 2 ( u ) + g 1 ( u )id , T 0 ( u ) = N Y i =1 [ u i + 1] q 1 / 2 [2] q 1 / 2 , g 1 ( u ) = N Y i =1 [ u i + 3] q 1 / 2 [ u i − 1] q 1 / 2 [2] 2 q 1 / 2 , (3.40) where u i = u − v i . W e ﬁr st consider the case a N = b N = a . Set L c,d = c d a b 1 b 2 b N − 1 b N u N − 1 u 1 − 1 · · · u N − 1 − 1 u N +1 u 1 +1 · · · u N − 1 + 1 • • • a a 1 a 2 a N − 1 a N (3.41) where each face stands for W = W 1 , 1 . T o the diﬀerence L a +1 ,d − L a − 1 ,d , one can apply the same trick as (3 .22). In particular, the rep eated use o f the star -triangle relation a nd the prop er ty W  b c a d     − 2  ∝ δ ac tells that it v anishes unless a i = b i for all i . Then the induction on N leads to the identit y L a +1 ,d −L a − 1 ,d = [2 a ] q 1 / 2 [2 a N ] q 1 / 2 N Y i =1 δ a i ,b i [ u i + 3] q 1 / 2 [ u i − 1] q 1 / 2 [2] 2 q 1 / 2 ! × ( 1 d = a N + 1 , − 1 d = a N − 1 . (3.42) 30 Now we are r eady to ev aluate the matrix e lement s of T 1 ( u + 1 ) T 1 ( u − 1 ). When a N = b N = a , we have ( T 1 ( u + 1) T 1 ( u − 1)) b 1 ,...,b N a 1 ,...,a N = L a − 1 ,a − 1 + L a +1 ,a +1 = L a − 1 ,a − 1 + L a − 1 ,a +1 + L a +1 ,a +1 − L a − 1 ,a +1 The ﬁr st tw o terms yield T 0 ( u ) T 2 ( u ) b 1 ,...,b N a 1 ,...,a N by the deﬁnition (3.23). The other t wo terms are e q ual to ( g 1 ( u )id) b 1 ,...,b N a 1 ,...,a N due to (3.42) with a = a N . When a N = b N ± 2, one can more easily chec k (3.40) since g 1 ( u )id do es not contribute. 3.7. V ertex and RSOS mo dels for general g . W e include a forma l and partly conjectural description of s olv able vertex and RSOS mo dels and their T-system for general g . W e will use the termino lo gy intro duced in later se c tions. (Therefore this techn ical section may b etter b e sk ipped on the ﬁrst reading.) Let W ( a ) m ( u ) b e the Kirillov-Reshetikhin mo dule (Section 4.2), where a ∈ I (set of vertices on the Dynkin diag ram of g ) a nd m ∈ Z ≥ 1 . It is a n irreducible ﬁnite dimensional represe n tation of unt wisted quantum a ﬃne algebr a U q = U q ( ˆ g ). Up to an ov er all scalar, there is the unique e lement , the R matr ix , R ∈ End( W ( a ) m ( u 1 ) ⊗ W ( b ) n ( u 2 )) characterized b y the in ter twining pro pe r ty ∆( U q ) P R = P R ∆( U q ), whe r e P is the tra nsp osition. It can in pr inciple b e co nstructed concr etely by solving this linear equation, or by the fusion of the simpler cas es m = n = 1 (cf. Theor em 4.3) or by taking the image of the univ er sal R . Let us denote the res ulting R ma trix by R ( a,m ; b,n ) ( z 1 /z 2 ), where z i = q tu i , t is deﬁned by (2.1 ) a nd the dependence thro ugh z 1 /z 2 is due to the genera l theor y . R ( a,m ; b,n ) ( z 1 /z 2 ) = W ( b ) n ( u 2 ) W ( a ) m ( u 1 ) z 1 /z 2 (3.43) As in (3.11), one introduces the r ow to ro w transfer matrix with the a ux iliary space W ( a ) m ( u ) by ( z = q tu ) T ( a ) m ( u ) = T r W ( a ) m ( u )  R ( a,m ; r N ,s N ) 0 ,N ( z / w N ) · · · R ( a,m ; r 1 ,s 1 ) 0 , 1 ( z / w 1 )  , ( 3.44) which acts on the quantum space W ( r 1 ) s 1 ( v 1 ) ⊗ · · · ⊗ W ( r N ) s N ( v N ) with w i = q tv i . They are a ll commutativ e, i.e. [ T ( a ) m ( u ) , T ( b ) n ( v )] = 0 thanks to the Y ang- B axter relatio n. It is a c orollar y o f the exa ct sequence under lying Theorem 4.8 and the argument on the c ent ral extension (cf. [1, section 2.2]) that T ( a ) m ( u ) satisﬁes the unrestricted T-system for g (2.22) with some scalars T ( a ) 0 ( u ) and g ( a ) m ( u ) appropria tely chosen depe nding on the normaliza tion o f T ( a ) m ( u ). Let ℓ ∈ Z ≥ 2 . F ro m the R matr ix one can in pr inciple construct the face Boltz- mann weight s for level ℓ U q ( ˆ g ) RSOS mo del a t q = exp  π √ − 1 t ( ℓ + h ∨ )  12 . Let us intro- duce P + = Z ≥ 0 ω 1 ⊕ · · · ⊕ Z ≥ 0 ω r , P ℓ = { λ ∈ P + | ( λ | maximal ro ot) ≤ ℓ } , (3.45) 12 Actually any primitive 2 t ( ℓ + h ∨ ) th ro ot of unity . h ∨ is the dual Coxet er num b er of g (2.3). 31 where ω a is a fundamental weigh t o f g (Section 2 .1). P ℓ is the cla ssical pro jectio n of the set o f level ℓ do minant integral weigh ts of the aﬃne Lie algebra ˆ g at level ℓ [11]. F or λ ∈ P + , let V λ be the irreducible U q ( g )-mo dule with highes t weigh t λ . Let res W ( a ) m be the (not necessa rily irreducible) U q ( g )-mo dule obtaine d b y re s tricting the U q ( ˆ g )-mo dule W ( a ) m ( u ). It is indep endent of u . See the text ar ound (4.22). When q is not a ro o t of unity , one has the irreducible deco mp os ition V λ ⊗ re s W ( a ) m ⊗ re s W ( b ) n = M µ ∈ P + Ω( λ ) µ ⊗ V µ , (3.46) where Ω( λ ) µ is the space of highest weight vectors of weight µ . Since ˇ R ( z ) = P R ( a,m ; b,n ) ( z ) comm utes with U q ( g ), the space Ω( λ ) µ is inv ariant under id ⊗ ˇ R ( z ). Thu s its matrix ele men ts yield the Bo ltzmann weigh ts o f unr estricted SOS mo del as in (3.29). The star- triangle relation for them follows from this constr uction. T o make the restr iction, we consider the ca se q = exp  π √ − 1 t ( ℓ + h ∨ )  , where the decomp osition (3.46) no longer holds [39, 40]. Howev er, based o n the obser v ation for g = A 1 [30], we conjecture that if λ is tak en from P ℓ and m ≤ t a ℓ, n ≤ t b ℓ , the quotient of the RHS of (3.46) by the t yp e I mo dules [41, 42] 13 reduces the sum µ ∈ P + to µ ∈ P ℓ , and id ⊗ ˇ R ( z ) remains well deﬁned on it. Then the RSOS Boltzmann weigh ts are deﬁned as the ma trix elemen ts o f id ⊗ ˇ R ( z ) on the quotient space, and satisfy the star-tria ngle relation. The RSOS mo del so constr ucted has the ﬂuctuating v ariable s on edges a s well as sites in general (cf. [4 3, Fig .1]). µ ν λ κ β γ δ α W ( a ) m W ( b ) n β ∈ Ω ( b,n ) µν γ ∈ Ω ( a,m ) λµ δ ∈ Ω ( a,m ) κν α ∈ Ω ( b,n ) λκ (3.47) The site v ariables b elong to P ℓ . In fact for g = A 1 , one may regar d the set of site v ariables { 1 , 2 , . . . , ℓ + 1 } as P ℓ = { 0 , ω 1 , . . . , ℓ ω 1 } . T o desc rib e the edge v aria bles, we consider the decomp osition V λ ⊗ res W ( a ) m = L µ ∈ P + Ω ( a,m ) λµ ⊗ V µ at gener ic q . When q = exp  π √ − 1 t ( ℓ + h ∨ )  , we need to take the quotient of the RHS by the type I modules, and this induces the quotient Ω ( a,m ) λµ of Ω ( a,m ) λµ . The e dge v ariable asso ciated to W ( a ) m belo ngs to the space Ω ( a,m ) λµ . W e set A ( a,m ) λµ = dim Ω ( a,m ) λµ and say that an (order ed) pair of site v ariables ( λ, µ ) ∈ P ℓ × P ℓ is admissible under W ( a ) m if A ( a,m ) λµ ≥ 1 14 . The matrix A ( a,m ) = ( A ( a,m ) λµ ) λ,µ ∈ P ℓ is called the admissibility ma trix of W ( a ) m . 13 Indecomposable mo dules with dim q = 0. See (14.49). 14 The type A r is bit sp ecial in that A ( a,m ) λµ ∈ { 0 , 1 } holds for any ( a, m ) and λ, µ , hence eﬀectiv ely no edge v ariable exists. Ho wev er , the situation A ( a,m ) λµ ≥ 2 still happens for the fusi on t yp es more general than those sp eciﬁed by Kirill o v- Reshetikhin mo dules [43]. 32 Let us formulate the row to row transfer ma trix T ( a ) m ( u ) that corresp o nds to the dual of the one for the vertex mo del (3.4 4). It acts on the spac e of paths H ( N ) = M λ i ∈ P ℓ Ω ( r 1 ,s 1 ) λ 1 λ 2 ⊗ · · · ⊗ Ω ( r N ,s N ) λ N λ 1 , (3.48) dim H ( N ) = T r  A ( r 1 ,s 1 ) · · · A ( r N ,s N )  . (3.49) The matrix elemen ts are depicted as follows ( u i = u − v i , λ i = λ i + N , µ i = µ i + N ): T ( a ) m ( u ) µ 1 ,β 1 ,µ 2 ,β 2 ,...,µ N ,β N λ 1 ,α 1 ,λ 2 ,α 2 ,...,λ N ,α N = X γ ∈ Ω ( a,m ) λ 1 µ 1 γ γ • • • µ 1 β 1 µ 2 β 2 µ 3 µ N β N µ 1 u 1 u 2 · · · u N λ 1 α 1 λ 2 α 2 λ 3 λ N α N λ 1 . (3.50) Here the symbols α i and β i denote a basis of Ω ( r i ,s i ) λ i λ i +1 and Ω ( r i ,s i ) µ i µ i +1 , resp ectively . The pairs ( λ i , λ i +1 ) and ( µ i , µ i +1 ) ar e b oth admis s ible under W ( r i ) s i , whereas ( λ i , µ i ) is so under W ( a ) m . The RHS stands for the pro duct o f the N Boltzmann weigh ts attached to the elementary sq uares summed o ver the states on the vertical edges accommo dating Ω ( a,m ) λ i µ i for i = 1 , . . . , N . As for the w eights, λ i +1 − λ i ≡ µ i +1 − µ i ≡ s i ω r i mo d the ro ot lattice; therefor e , the T ( a ) m ( u ) under c onsideration is v anishing unless N X i =1 s i  C − 1  a r i ∈ Z for all a ∈ I , (3.51) where C is the Cartan matrix o f g (Section 2.1). Due to the star-triang le rela- tion (including sums ov er edg e v ariables), the comm utativity [ T ( a ) m ( u ) , T ( b ) n ( v )] = 0 holds. W e conjectur e that T ( a ) m ( u ) satisﬁes the level ℓ res tricted T- s ystem for g o f the form (2.22) with some sca lars T ( a ) 0 ( u ) and g ( a ) m ( u ) appro priately ch osen dep end- ing on the nor malization. In particular , this implies that the | P ℓ | by | P ℓ | matric e s A ( a,m ) with a ∈ I , 0 ≤ m ≤ t a ℓ a re c o mmu tative and sa tisfy the level ℓ restricted Q-system (cf. Section 14.5) with the b ounda ry condition A ( a, 1) = 1 , A ( a,t a ℓ +1) = 0 15 . (3.52) Let dim q V λ be the q -dimension o f V λ at q = exp  π √ − 1 t ( ℓ + h ∨ )  deﬁned in (14.49). W e set Q ( a ) m = dim q res W ( a ) m , which suppo sedly satisﬁes the level ℓ restr icted Q -system (14.5) (Conjecture 14.2). Now the generalization of (3.37) is given as Q ( a ) m dim q V λ = X µ ∈ P ℓ A ( a,m ) λµ dim q V µ ( λ ∈ P ℓ ) . (3.53) Since dim q V λ > 0 for a ny λ ∈ P ℓ , the Perron-F rob e nius theorem tells that Q ( a ) m is the larg est eigenv alue of the a dmissibility matr ix A ( a,m ) . Therefore in the homo- geneous case where ( r i , s i ) = ( p, s ) for all i , we ﬁnd from (3.49) that lim N →∞ (dim H ( N )) 1 / N = Q ( p ) s . (3.54) 15 This leads to Q b ∈ I ( A ( b,t b ℓ ) ) C ab = 1 f or an y a ∈ I , which is a we ak er constrain t than A ( a,t a ℓ ) = 1 employ ed in the deﬁnition of the level ℓ r estricted Q-system in Section 14.5. 33 This prop erty will b e re-der ived in the TBA analys is in (1 5.20). In general, the Boltzmann weights (3.47) are express ed in ter ms of the function [ u ] q t/ 2 ∝ sin  π u 2( ℓ + h ∨ )  . ( t is deﬁned in (2.1).) This is indeed the cas e for A 1 as in (3.17) and in the o ther known ex amples. It is als o consistent with the Bethe equa- tion (8.2 5). Consequently , the tra nsfer matrix with an a ppropriate no rmalization po ssesses the p erio dicity T ( a ) m ( u + 2( ℓ + h ∨ )) = T ( a ) m ( u ) . (3.55 ) W e will see in Theo rem 5.7 that the level ℓ restric ted T-system in Section 2.2 16 alone compels this prop erty . 3.8. Bibliog raphical notes. T he in tegr ability of the 6 vertex mo del (3.1) (ﬁrs t solved in [25, 26]) has b een for m ulated in terms of the Y ang-Baxter eq uation and commuting tra nsfer matrices in [2]. So lutio ns of the Y a ng-Baxter equation that hav e bee n k nown by 19 80 ar e survey ed in [4 4] fro m the p ersp ective of the q uantum in verse scattering metho d. Subsequent g eneralizatio ns of trigonometric vertex mo dels to t yp e A [45, 4 6, 47] and ma n y other g [4 8, 49] hav e b een ass em bled in the repr int volume [50]. The fusion o f vertex mo dels is for mulated in [27]. See also [51]. The idea of utilizing the functional relatio ns of transfer matrices go es back to B axter [52, 2]. Some simplest exa mples o f the T-sys tem hav e b een obta ine d for the XXZ chain [5 3], the O ( n )-symmetry mo dels [54] a nd vertex mo dels asso cia ted with some other g [55]. With reg ard to the RSOS mo dels, the 8VSOS mo de l is the fundamental example containing the Ising a nd (generalized) har d hexagon mo dels as the le vel ℓ = 2 , 3 cases, resp ectively . The one po int function [34] essentially gives rise to the charac- ter of the Viraso ro minimal series, and this fact inspir e d intensive studies o n the relations with conformal ﬁeld theory and represe ntation theo r y of quantum aﬃne algebras . In the termino lo gy in Section 3 .7, the 8VSOS mo del cor r esp onds to the level ℓ RSOS mo del for g = A 1 with fusion type W (1) 1 (bo th on the ho rizontal and vertical edges). Beyond the A 1 case, concrete constr uctions of RSOS mo dels for unt wisted aﬃne Lie alg ebra ˆ g hav e be en done for non exceptional serie s g = A r , B r , C r , D r [56, 57] asso ciated with W (1) 1 (“vector representation”) and g = G 2 [58] with W (2) 1 . The fusion of RSOS mo dels hav e b e en w orked o ut ex plicitly only for type A [35, 43]. One of the earlies t examples of the T-system for RSOS mo dels (except the Ising) is [36] for the generalized ha rd hexa gon mo del. It was s ystematized to the general level res tr icted T-sys tem for A 1 in [7 ]. See also [37] where the rela tion of the for m “ T m T 1 = T m − 1 + T m +1 ” was given. In [59], the Jacobi-T rudi t yp e functional relations (cf. Theorem 6.1 and 6.2) were given for the fusion RSOS mo dels of type A r . The T-system for A r is extracted from them in [1], where the extension to all g was prop os ed base d on the co nnection to the Y-system and the Q-sys tem. Finally , one ca n construct the quan tum ﬁeld theor y analog of the commuting tr a nsfer matrices that act on Viraso r o F o ck spaces and s atisfy the T- system. See [60] for the or iginal c o nstruction fo r g = A 1 and [61] for a recent application. 16 In this case, the normalization i s T ( a ) 0 ( u ) = T ( a ) t a ℓ ( u ) = 1. 34 4. T-system in q uantum group theor y 4.1. Quan tum aﬃ ne algebra. F or simplicity w e concentrate on the unt wisted quantum aﬃne a lg ebra U q ( ˆ g ) un til Sectio n 4 .5. W e a s sume that q ∈ C × is not a ro ot of unity and set q = e ~ ; therefore, the domain U of the sp ectral para meter u should b e understo o d as U = C t ~ . See Sectio n 2.1. W e set ˆ I = { 0 } ⊔ I and let ˆ C = ( ˆ C ij ) i,j ∈ ˆ I be the Ca rtan matrix of the aﬃne Ka c-Mo o dy algebra ˆ g [11]. F or i, j ∈ I , o ne has ˆ C ij = C ij where the latter is an element of the Ca rtan ma trix C of g . By deﬁnition, the (un twisted) quantum aﬃne algebra U q ( ˆ g ) [6 2, 14] is the asso ciative algebr a ov e r C with gener ators x ± i , k ± 1 i , ( i ∈ ˆ I ) and the relatio ns: k i k − 1 i = k − 1 i k i = 1 , k i k j = k j k i , k i x ± j k − 1 i = q ± ˆ C ij i x ± j , [ x + i , x − j ] = δ ij k i − k − 1 i q i − q − 1 i , 1 − ˆ C ij X ν =0 ( − 1) ν " 1 − ˆ C ij ν # q i ( x ± i ) 1 − ˆ C ij − ν x ± j ( x ± i ) ν = 0 ( i 6 = j ) . (4.1) Here q 0 = q a nd q i = q t/t i for i ∈ I . F or the notations t and t i , s e e (2.1). F urthermor e, for 0 ≤ n ≤ m , h m n i q = [ m ] q ! [ n ] q ![ m − n ] q ! , [ m ] q ! = [1 ] q [2] q · · · [ m ] q . (4.2) See (3.18) for the deﬁnition o f [ m ] q . The algebr a U q ( ˆ g ) is denoted by U ′ q ( ˆ g ) in some literature indicating that the analo g of the deriv ation op era tor in ˆ g has no t b een included. There are 2 r +1 algebra automorphisms of U q ( ˆ g ) given o n gener ators b y k i 7→ σ i k i , x + i 7→ σ i x + i , x − i 7→ x − i (4.3) for an y set o f signs σ 0 , . . . , σ r ∈ {± 1 } . Ob v io usly , U q ( ˆ g ) contains U q ( g ) as a subal- gebra. There is another r ealization of U q ( ˆ g ) called the Dr infeld new realiza tion [63, 64]. Namely , U q ( ˆ g ) is is omorphic to the algebra with g enerators x ± i,n ( i ∈ I , n ∈ Z ), k ± 1 i ( i ∈ I ), h i,n ( i ∈ I , n ∈ Z \{ 0 } ) and central element s c ± 1 / 2 , with the following relations: k i k j = k j k i , k i h j,n = h j,n k i , k i x ± j,n k − 1 i = q ± C ij i x ± j,n , [ h i,n , x ± j,m ] = ± 1 n [ nC ij ] q i c ∓| n | / 2 x ± j,n + m , [ h i,n , h j,m ] = δ n, − m 1 n [ nC ij ] q i c n − c − n q j − q − 1 j , x ± i,n +1 x ± j,m − q ± C ij i x ± j,m x ± i,n +1 = q ± C ij i x ± i,n x ± j,m +1 − x ± j,m +1 x ± i,n , [ x + i,n , x − j,m ] = δ ij c ( n − m ) / 2 φ + i,n + m − c − ( n − m ) / 2 φ − i,n + m q i − q − 1 i , X π ∈ Σ s s X k =0 ( − 1) k h s k i q i x ± i,n π (1) . . . x ± i,n π ( k ) x ± j,m x ± i,n π ( k +1) . . . x ± i,n π ( s ) = 0 , i 6 = j (4.4) 35 for all sequences of in tege rs n 1 , . . . , n s , wher e s = 1 − C ij , Σ s is the symmetric group on s letters, and φ ± i,n ’s are determined by the for mal power series ∞ X n =0 φ ± i, ± n ζ ± n = k ± 1 i exp ± ( q i − q − 1 i ) ∞ X m =1 h i, ± m ζ ± m ! . (4.5) In the tw o realiza tio ns (4.1) and (4.4), the symbol k ± 1 i ( i ∈ I ) stands for the s ame generator under the isomorphism. U q ( ˆ g ) admits a Hopf algebra structure [62, 14]. 4.2. Finite dimens ional representat ions. A representation W of U q ( ˆ g ) is called typ e 1 if the generator s k 0 , k 1 , . . . , k r act semi simply o n W with eigenv a lues in q Z and c 1 / 2 in (4.4) acts as 1 on W . A vector v ∈ W is called a highest w e ig ht vector if x + i,n · v = 0 , φ ± i,n · v = ψ ± i,n v , c 1 / 2 v = v , (4.6) for some complex num ber s ψ ± i,n . A type 1 representation W is called a highest weigh t representation if W = U q ( ˆ g ) · v for some highest weigh t vector v . Theorem 4.1 ([65, 66]) . (1) Every ﬁ n ite dimensional irr e ducible r epr esentation of U q ( ˆ g ) c an b e obtaine d fr om a typ e 1 r epr esent ation by a t wisting with an automor- phism (4.3 ). (2) Every ﬁnite dimensional irr e ducible re pr esentation of U q ( ˆ g ) of typ e 1 is a highest weight r epr esent ation. (3) A typ e 1 highest weight r epr esentation with the highest weight ve ctor v in (4.6) is ﬁnite dimensional if and only if t her e exist p olynomials P a ( ζ ) ∈ C [ ζ ] ( a ∈ I ) such that P a (0) = 1 and X n ≥ 0 ψ ± a, ± n ζ ± n = q deg P a a P a ( ζ q − 1 a ) P a ( ζ q a ) ∈ C [[ ζ ± 1 ]] . (4.7) The p olyno mials P a ( ζ ) are ca lled Drinfeld po ly nomials after the a nalogous cla s- siﬁcation theorem by Drinfeld for Y a ngians [63]. The Kir illov-Reshetikhin mo dule W ( a ) m ( u ) ( a ∈ I , m ∈ Z ≥ 1 , u ∈ C t ~ ) is the ir re- ducible ﬁnite dimensio nal r epresentation of U q ( ˆ g ) that co rresp onds to the Drinfeld po lynomial P b ( ζ ) = ( Q m s =1 (1 − ζ q tu q m +1 − 2 s a ) if b = a, 1 otherwise . (4.8) This W ( a ) m ( u ) is equal to W ( a ) m,q tu q − m +1 i in [6 7, 68]. In particular , W (1) 1 ( u ) , . . . , W ( r ) 1 ( u ) are called fundamen tal representations. 4.3. Example. Co nsider the s implest example U q = U q ( A (1) 1 ). In realiza tion (4.1), ˆ I = { 0 , 1 } and the Cartan matrix is ˆ C =  2 − 2 − 2 2  . The copro duct is given b y ∆ x + i = x + i ⊗ 1 + k i ⊗ x + i , ∆ x − i = x − i ⊗ k − 1 i + 1 ⊗ x − i , ∆ k ± 1 i = k ± 1 i ⊗ k ± 1 i . (4.9) F or m ∈ Z ≥ 0 , let W m ( u ) = W (1) m ( u ) b e the K irillov-Reshetikhin mo dule. Plainly , it is the m + 1 dimensional (i.e. spin m 2 ) ir reducible representation W m ( u ) = 36 C v m 1 ⊕ · · · ⊕ C v m m +1 given by ( z = q u ) x − 1 v m j = [ m + 1 − j ] v m j +1 , x + 1 v m j = [ j − 1] v m j − 1 , k ± 1 1 v m j = q ± ( m +2 − 2 j ) v m j , (4.10) x + 0 v m j = z [ m + 1 − j ] v m j +1 , x − 0 v m j = z − 1 [ j − 1] v m j − 1 , k ± 1 0 v m j = q ∓ ( m +2 − 2 j ) v m j , (4.11) where [ j ] = [ j ] q = q j − q − j q − q − 1 as in (3.18). In the Drinfeld new re alization (4.4), the highest weigh t vector is identiﬁed with v m 1 and the eigenv alues in (4 .6) r ead ψ ± 1 , ± n = ( q ± m n = 0 , ± ( q m − q − m )( z q m ) ± n n ≥ 1 . The relation (4.7) holds with the Drinfeld p olyno mial P 1 ( ζ ) = (1 − ζ q u − m +1 )(1 − ζ q u − m +3 ) · · · (1 − ζ q u + m − 1 ) in agreement with (4.8). The exa ct sequence (3.16) is reﬁned along the deﬁnitions her e. The vectors v i ∈ V 1 and x = ( x 1 , x 2 ) ∈ V m in Section 3.1 are to be identiﬁed with v 1 i and v m x 2 +1 in (4.10)–(4.11), resp ectively . W e introduce the base of W 1 ( u ) ⊗ W 1 ( v ) as u 1 = v 1 1 ⊗ v 1 1 , u 2 = 1 [2] ∆( x − 1 ) u 1 = v 1 1 ⊗ v 1 2 + q − 1 v 1 2 ⊗ v 1 1 [2] , u ′ 1 = v 1 1 ⊗ v 1 2 − q v 1 2 ⊗ v 1 1 , u 3 = ∆( x − 1 ) u 2 = v 1 2 ⊗ v 1 2 . (4.12) Under the action of x ± 1 , k ± 1 1 , the set o f vectors { u 1 , u 2 , u 3 } and { u ′ 1 } b ehav e a s the triplet and the singlet represe ntations as usual. On the o ther hand, with r egard to x ± 0 , they are mixed as follows. ( x = q u , y = q v ) ∆( x + 0 ) : ✲ ∆( x − 0 ) : ✲ u 1 ✻ ❄ ( x − 1 + y − 1 )[2] − 1 x + y ❥ ❨ y − 1 − q 2 x − 1 ( y q − 2 − x )[2] − 1 u 2 u ′ 1 ( x + y )[2] − 1 x − 1 + y − 1 ✯ ✙ x − q 2 y ( q − 2 x − 1 − y − 1 )[2] − 1 ✻ ❄ u 3 (4.13) The diag ram means ∆( x + 0 ) u 1 = ( x + y ) u 2 + y q − 2 − x [2] u ′ 1 for instance. F rom (4.13), we ﬁnd that W 1 ( u ) ⊗ W 1 ( v ) is irreducible if a nd only if x y 6 = q ± 2 , namely u − v 6 = ± 2. 37 In the reducible cases, (4.13) lo oks as ∆( x + 0 ) : ✲ ∆( x − 0 ) : ✲ u 1 ✻ ❄ z − 1 z [2] ■ u 2 u ′ 1 z z − 1 [2] ✠ ✻ ❄ u 3 (i) z := q x = q − 1 y u 1 ✻ ❄ z − 1 z [2] ❘ u 2 u ′ 1 z z − 1 [2] ✒ ✻ ❄ u 3 (ii) z := q − 1 x = q y . (4.14) In the bo th ca ses, W 1 ( u ) ⊗ W 1 ( v ) is indecomp osable and the subspace C u 1 ⊕ C u 2 ⊕ C u 3 bec omes is omorphic to W 2 ( u + v 2 ) corres p o nding to the m ultiplicative sp ectral parameter z . The diﬀerence is tha t W 2 ( u + v 2 ) is the irreducible submo dule in the case o f (i) while it is the irreducible q uo tient for (ii). Denoting the trivial one dimensional mo dule C u ′ 1 by W 0 , w e th us get the exact sequences of U q -mo dules: (i) 0 → W 2 ( u ) → W 1 ( u − 1) ⊗ W 1 ( u + 1) → W 0 → 0 , (4.15) (ii) 0 → W 0 → W 1 ( u + 1) ⊗ W 1 ( u − 1) → W 2 ( u ) → 0 . (4.16) The ge neral case, which was ﬁrs t w o r ked out in the context of Y ang ian, is summa- rized in Theorem 4 . 2 ([6 9]) . W m ( u ) ⊗ W n ( v ) is r e ducible if and only if | u − v | = m + n − 2 j + 2 for some 1 ≤ j ≤ min( m, n ) . In these c ase, the fol lowing exact se quenc es ar e valid: 0 → W j − 1 ( u + m − j + 1) ⊗ W m + n − j +1 ( v − m + j − 1) → W m ( u ) ⊗ W n ( v ) → W m − j ( u − j ) ⊗ W n − j ( v + j ) → 0 for v − u = m + n − 2 j + 2 . 0 → W m − j ( u + j ) ⊗ W n − j ( v − j ) → W m ( u ) ⊗ W n ( v ) → W j − 1 ( u − m + j − 1) ⊗ W m + n − j +1 ( v + m − j + 1) → 0 (4.17) for u − v = m + n − 2 j + 2 . 4.4. q -c haracters. Let Rep U q ( ˆ g ) be the Grothendieck ring of the catego r y of the t yp e 1 ﬁnite dimensional U q ( ˆ g )-mo dules. Such a mo dule W allows the direct sum decomp osition W = M γ =( γ ± a, ± n ) i ∈ I ,n ≥ 0 W γ , W γ = { v ∈ W | ∃ p ≥ 0 , ∀ a ∈ I , n ≥ 0 , ( φ ± a, ± n − γ ± a, ± n ) p v = 0 } . It can b e shown [70] that the generating function of the (genera lized) eigenv a lues is expressed as X n> 0 γ ± a, ± n ζ ± n = q deg R + a − deg R − a a R + a ( ζ q − 1 a ) R − a ( ζ q a ) R + a ( ζ q a ) R − a ( ζ q − 1 a ) ∈ C [[ ζ ± 1 ]] (4.18 ) in terms of some po lynomials R ± a ( ζ ) in ζ with constant term 1. 38 Let Z [ Y ± 1 a,z ] a ∈ I ,z ∈ C × be the ring of integer co e ﬃcie n t Laur e n t po lynomials in inﬁnitely many alg ebraically indep endent v aria bles { Y a,z | a ∈ I , z ∈ C × } 17 . The F renkel-Reshetikhin q -character χ q is the injectiv e ring morphism χ q : Rep U q ( ˆ g ) → Z [ Y ± 1 a,z ] a ∈ I ,z ∈ C × , χ q ( W ) = X γ dim( W γ ) m γ , (4.19) where the monomial m γ is spec iﬁe d from R ± a ( ζ ) (4 .18) by m γ = Y a ∈ I ,z ∈ C × Y r + a,z − r − a,z a,z , R ± a ( ζ ) = Y z ∈ C × (1 − ζ z ) r ± a,z . (4.20) Suppo se that W is the irr educible repr esentation with Dr infeld p oly no mials P a ( ζ ) = Q m a s =1 (1 − ζ z ( a ) s ). Compa ring (4.7) with (4.18) a nd (4.20), o ne ﬁnds that its q -character χ q ( W ) contains the monomial Q r a =1 Q m a s =1 Y a,z ( a ) s corres p o nding to the highest weigh t vector. Such a monomia l is ca lled a hig hest weigh t monomial. Thu s, in particula r , the q -character of the K irillov-Reshetikhin module W ( a ) m ( u ) is a Laurent p olynomia l co ntaining the highest weigh t monomial as χ q ( W ( a ) m ( u )) = m Y s =1 Y a,z q m +1 − 2 s a + · · · , (4.21) where we hav e set z = q tu . The ca se m = 1 is ca lled the fundamental q - character. F or an analog ous trea tmen t of the Y angians, see [71]. Deﬁne Ch U q ( ˆ g ) to be the image Im χ q and call it the q -character ring of U q ( ˆ g ). By the deﬁnitio n, Ch U q ( ˆ g ) is an integral domain and a commutativ e r ing isomor - phic to Rep U q ( ˆ g ). The following fact is well known. Theorem 4.3 ([70], Cor ollary 2 ) . The ring Ch U q ( ˆ g ) is fr e ely gener ate d by t he fundamental q -char acters χ q ( W ( a ) 1 ( u )) ( a ∈ I , u ∈ U ). Example 4 .4. F or g = A 1 , the q -character of the Kirillov-Reshetikhin mo dule W (1) m ( u ) is given by ( z = q u , Y z = Y 1 ,z ) χ q ( W (1) 1 ( u )) = Y z + Y − 1 z q 2 , χ q ( W (1) 2 ( u )) = Y z q − 1 Y z q + Y z q − 1 Y − 1 z q 3 + Y − 1 z q Y − 1 z q 3 , and, in genera l , χ q ( W (1) m ( u )) = m X j =0 m − j Y i =1 Y z q − m − 1+2 i j Y k =1 Y − 1 z q m +3 − 2 k . 17 The v ari able Y a,z is unrelated to the Y of Y-systems. 39 Example 4.5. W e write down the fundamental q -characters χ q ( W ( a ) 1 ( u )) for g with rank 2 ( z = q tu ). A 2 : χ q ( W (1) 1 ( u )) = Y 1 ,z + Y − 1 1 ,z q 2 Y 2 ,z q + Y − 1 2 ,z q 3 , χ q ( W (2) 1 ( u )) = Y 2 ,z + Y 1 ,z q Y − 1 2 ,z q 2 + Y − 1 1 ,z q 3 , B 2 : χ q ( W (1) 1 ( u )) = Y 1 ,z + Y − 1 1 ,z q 4 Y 2 ,z q Y 2 ,z q 3 + Y 2 ,z q Y − 1 2 ,z q 5 + Y 1 ,z q 2 Y − 1 2 ,z q 3 Y − 1 2 ,z q 5 + Y − 1 1 ,z q 6 , χ q ( W (2) 1 ( u )) = Y 2 ,z + Y 1 ,z q Y − 1 2 ,z q 2 + Y − 1 1 ,z q 5 Y 2 ,z q 4 + Y − 1 2 ,z q 6 , C 2 : χ q ( W ( a ) 1 ( u )) = ( χ q ( W (3 − a ) 1 ( u )) for B 2 ) | Y 1 ,z ↔ Y 2 ,z ( a = 1 , 2) , G 2 : χ q ( W (1) 1 ( u )) = Y 1 ,z + Y 2 ,z q Y 2 ,z q 3 Y 2 ,z q 5 Y − 1 1 ,z q 6 + Y 2 ,z q Y 2 ,z q 3 Y − 1 2 ,z q 7 + Y 1 ,z q 4 Y 2 ,z q Y − 1 2 ,z q 5 Y − 1 2 ,z q 7 + Y 1 ,z q 2 Y 1 ,z q 4 Y − 1 2 ,z q 3 Y − 1 2 ,z q 5 Y − 1 2 ,z q 7 + Y 2 ,z q Y 2 ,z q 9 Y − 1 1 ,z q 10 + Y 1 ,z q 2 Y 2 ,z q 9 Y − 1 1 ,z q 10 Y − 1 2 ,z q 3 + Y 2 ,z q Y − 1 2 ,z q 11 + Y 1 ,z q 4 Y − 1 1 ,z q 8 + Y 2 ,z q 5 Y 2 ,z q 7 Y 2 ,z q 9 Y − 1 1 ,z q 8 Y − 1 1 ,z q 10 + Y 1 ,z q 2 Y − 1 2 ,z q 3 Y − 1 2 ,z q 11 + Y 2 ,z q 5 Y 2 ,z q 7 Y − 1 1 ,z q 8 Y − 1 2 ,z q 11 + Y 2 ,z q 5 Y − 1 2 ,z q 9 Y − 1 2 ,z q 11 + Y 1 ,z q 6 Y − 1 2 ,z q 7 Y − 1 2 ,z q 9 Y − 1 2 ,z q 11 + Y − 1 1 ,z q 12 , χ q ( W (2) 1 ( u )) = Y 2 ,z + Y 1 ,z q Y − 1 2 ,z q 2 + Y − 1 1 ,z q 7 Y 2 ,z q 4 Y 2 ,z q 6 + Y 2 ,z q 4 Y − 1 2 ,z q 8 + Y 1 ,z q 5 Y − 1 2 ,z q 6 Y − 1 2 ,z q 8 + Y − 1 1 ,z q 11 Y 2 ,z q 10 + Y − 1 2 ,z q 12 . More examples will b e given in Sectio ns 7.1 – 7.4. An y ﬁnite dimensional U q ( ˆ g )-mo dule W deﬁnes a r epresentation of the subalg e - bra U q ( g ), which we denote b y res W . The (usual) character χ o f the la tter lives in Z [ y ± 1 a ] a ∈ I with y a = e ω a with ω a being a fundamental weigh t. The q -character is a deformation of the character by z in that res χ q ( W ) = χ (res W ) , (4.22) where res on the LHS is to b e understo o d as res : Z [ Y ± 1 a,z ] a ∈ I ,z ∈ C × → Z [ y ± 1 a ] a ∈ I Y a,z 7→ y a . (4.23) Note that r es W is no t necessarily an irreducible U q ( g )-mo dule even if W is s o as a U q ( ˆ g )-mo dule. Therefor e the irreducible q - character χ q ( W ( a ) m ( u )) do es not restrict to an irreducible character in g eneral. In fact in Exa mple 4.5, o ne observes res χ q ( W (1) 1 ( u )) = ( χ ( V ω 1 ) + χ ( V 0 ) if g = G 2 and a = 1 , χ ( V ω a ) otherwise , (4.24) where V λ denotes the irr educible U q ( g )-mo dule with hig hest weigh t λ . The a lgebra g = A r is e xceptional in that r e s χ q ( W ( a ) m ( u )) = χ ( V mω a ) holds for a ll a and m . See (7.7) a nd (13.63). A systematic treatment of such deco mpo sitions is r elated to the Kiril lov-R eshetikhin c onje ctu re which has b een fully solved by now. See Section 13, esp ecially Section 13.7. F or a ∈ I and z ∈ C × , set A a,z = Y a,z q − 1 a Y a,z q a Y b : C ba = − 1 Y − 1 b,z Y b : C ba = − 2 Y − 1 b,z q − 1 Y − 1 b,z q Y b : C ba = − 3 Y − 1 b,z q − 2 Y − 1 b,z Y − 1 b,z q 2 . (4.25) 40 By the deﬁnition, o ne has r e s A a,z = Q b ∈ I y C ba b = e α a with α a being a simple ro o t. Let S a ( a ∈ I ) b e the scr eening op era tor [70]. Namely , S a sends Z [ Y ± 1 a,z ] a ∈ I ,z ∈ C × to the extended ring adjoined with the extra symbols S a,z with a ∈ I , z ∈ C × . The action is given by S a · Y b,z = δ ab Y a,z S a,z (4.26) and the Leibniz rule S a · ( Y Z ) = ( S a · Y ) Z + Y ( S a · Z ). Th us for example, S a · Y − 1 b,z = − δ ab Y − 1 a,z S a,z . The symbo l S a,z is assumed to ob ey the re lation S a,z q 2 a = A a,z q a S a,z (4.27) in the extended ring. Theorem 4.6 ([70, 72]) . (1) The q -char acter of an irr e ducible ﬁ nite dimensional U q ( ˆ g ) -mo dule W has the form χ q ( W ) = m + (1 + P p M p ) , wher e m + is t he highest weight monomial and e ach M p is a monomial in A − 1 a,z , a ∈ I , z ∈ C × , (i.e. it do es not c ont ain any p ositive p ower factors of A a,z ). (2) The image Im χ q ( ≃ Ch U q ( ˆ g )) of t he q - char acter morphism (4.19) is e qual to T r a =1 Ker S a . The assertio n (1 ) is a natura l analog of its undefor med version res χ q ( W ) ∈ res( m + )(1 + P α Z ≥ 0 e − α ), where res( m + ) = e highest w e ight and the α -sum runs ov er Z ≥ 0 α 1 ⊕ · · · ⊕ Z ≥ 0 α r \ { 0 } . The assertion (2) has a background in the characterization of the (deformed) W -a lgebra a s the int ersection of the kernel o f screening op erator s [70]. Example 4. 7. Let us illustrate Theore m 4.6 along g = A 2 . The deﬁnition (4.25) reads A 1 ,z = Y 1 ,z q − 1 Y 1 ,z q Y − 1 2 ,z , A 2 ,z = Y 2 ,z q − 1 Y 2 ,z q Y − 1 1 ,z . T ake χ q = χ q ( W (1) 1 ( u )) = Y 1 ,z + Y − 1 1 ,z q 2 Y 2 ,z q + Y − 1 2 ,z q 3 for A 2 in Example 4.5. The highest weigh t monomial is Y 1 ,z . χ q is expressed as χ q = Y 1 ,q (1 + A − 1 1 ,z q + A − 1 1 ,z q A − 1 2 ,z q 2 ) (4.28) in agr eement with (1). With regar d to (2), let us check that χ q belo ngs to Ker S 1 T Ker S 2 . S 1 · χ q = Y 1 ,z S 1 ,z − Y − 1 1 ,z q 2 Y 2 ,z q S 1 ,z q 2 = Y 1 ,z S 1 ,z − Y − 1 1 ,z q 2 Y 2 ,z q A 1 ,z q S 1 ,z = 0 , S 2 · χ q = Y − 1 1 ,z q 2 Y 2 ,z q S 2 ,z q − Y − 1 2 ,z q 3 S 2 ,z q 3 = Y − 1 1 ,z q 2 Y 2 ,z q S 2 ,z q − Y − 1 2 ,z q 3 A 2 ,z q 2 S 2 ,z q = 0 . 4.5. T-system and q -cha racters. W e co nt inu e to set u ∈ U = C t ~ in this subsec- tion. The following is the fundamental result that relates the Kir illov-Reshetikhin mo dules with the T-system. Theorem 4.8 ([67, 6 8]) . F or any g , T ( a ) m ( u ) = χ q ( W ( a ) m ( u )) satisﬁes the un r e- stricte d T-system for g . In fact, the exact s equence corr esp onding to the g -version o f j = n = m case of (4.17) has b een obtained. It is a n elementary ex ercise to chec k that the q -characters for g = A 1 in Example 4.4 satisfy the T-s y stem (2.6). Theorem 4 .8 lea ds to a description of the ring Rep U q ( ˆ g ) ≃ Ch U q ( ˆ g ) by the q -characters of the K ir illov-Reshetikhin modules and the unrestricted T-system, which we sha ll now explain. Let T = { T ( a ) m ( u ) | a ∈ I , m ∈ Z ≥ 1 , u ∈ U } deno te the family o f v ariables. Let T ( g ) b e the r ing with generator s T ( a ) m ( u ) ± 1 with the 41 relations given b y the T-system for g . Deﬁne T ◦ ( g ) to b e the subr ing of T ( g ) generated b y T . Theorem 4 . 9 ([17]) . The ring T ◦ ( g ) is isomorphic to Rep U q ( ˆ g ) by the c orr esp on- denc e T ( a ) m ( u ) 7→ W ( a ) m ( u ) . 4.6. T-system for quan tum aﬃnizations of quan tum Kac-Mo o dy alge- bras. The T-systems hav e b een gener alized by Herna ndez [8] to the quantum aﬃnizations of a wide class of quan tum Kac– Mo o dy alge bras s tudied in [63, 73, 74, 75, 76, 77]. The most dis tinct fea tur e compared from the s etting so far is that the categ ory Rep U q ( ˆ g ) and the tenso r pro duct ⊗ need to b e replace d by Mo d( U q ( ˆ g )) co nsisting of no t necessa r ily ﬁnite dimensional mo dules and the fu- sion pro duct ∗ f , r esp ectively . Nevertheless, with an appro priate deﬁnition of the Kirillov-Reshetikhin mo dules a nd their q - characters, the la tter satisfy the (gener- alized) T-system [8]. Here w e o nly give the deﬁnition of the quantum aﬃniza tion of quan tum Ka c- Mo o dy alg ebras and write down the T-sys tem, leaving many deta ils to [8]. Instead, we include the explicit form of the co rresp onding Y-system [78] on which our pre- sentation is mainly based. W e b egin b y resetting the deﬁnitions and no tations such as C, t, q i , g , ˆ g , U q ( g ) and U q ( ˆ g ) in tr o duced so far 18 . Let I = { 1 , . . . , r } and let C = ( C ij ) i,j ∈ I be a gener alize d Cartan matrix in [11]; na mely , it s atisﬁes C ij ∈ Z , C ii = 2, C ij ≤ 0 for a ny i 6 = j , and C ij = 0 if and only if C j i = 0. W e as s ume that C is symmetrizable , i.e. there is a diagonal matrix D = diag( d 1 , . . . , d r ) with d i ∈ Z ≥ 1 such that B = D C is symmetric. W e assume that there is no common divisor for d 1 , . . . , d r except for 1. Let ( h , Π , Π ∨ ) b e a realizatio n of the Car ta n matrix C [11]; namely , h is a (2 r − rank C ) dimensio nal Q -vector spa ce, and Π = { α 1 , . . . , α r } ⊂ h ∗ , Π ∨ = { α ∨ 1 , . . . , α ∨ r } ⊂ h such that α j ( α ∨ i ) = C ij . Let q ∈ C × be not a ro ot o f unity . W e set q i = q d i ( i ∈ I ) and us e the s ymbols deﬁned in (4.2). Let U q ( g ) b e the quan- tum Kac- Mo o dy algebra [62, 14], which is a q -a na log of the Kac - Mo o dy algebra g asso ciated with C [1 1]. The quantum aﬃnization (without c e n tral ele ments) o f the quantum Kac– Mo o dy algebra U q ( g ), deno ted by U q ( ˆ g ), is the C -algebr a with gene r ators x ± i,n ( i ∈ I , 18 This reset is only for the curr ent subsection. 42 n ∈ Z ), k h ( h ∈ h ), h i,n ( i ∈ I , n ∈ Z \ { 0 } ) and the following r elations: k h k h ′ = k h + h ′ , k 0 = 1 , k h φ ± i ( z ) = φ ± i ( z ) k h , k h x ± i ( z ) = q ± α i ( h ) x ± i ( z ) k h , φ + i ( z ) x ± j ( w ) = q ± B ij w − z w − q ± B ij z x ± j ( w ) φ + i ( z ) , φ − i ( z ) x ± j ( w ) = q ± B ij w − z w − q ± B ij z x ± j ( w ) φ − i ( z ) , x + i ( z ) x − j ( w ) − x − j ( w ) x + i ( z ) = δ ij q i − q − 1 i  δ  w z  φ + i ( w ) − δ  z w  φ − i ( z )  , ( w − q ± B ij z ) x ± i ( z ) x ± j ( w ) = ( q ± B ij w − z ) x ± j ( w ) x ± i ( z ) , X π ∈ Σ 1 − C ij X k =1 ( − 1) k  1 − C ij k  q i x ± i ( w π (1) ) · · · x ± i ( w π ( k ) ) x ± j ( z ) × x ± i ( w π ( k +1) ) · · · x ± i ( w π (1 − C ij ) ) = 0 ( i 6 = j ) . (4.29) In (4.29) Σ is the symmetric group for the set { 1 , . . . , 1 − C ij } . W e hav e also used the following formal ser ie s : x ± i ( z ) = X n ∈ Z x ± i,n z n , φ ± i ( z ) = k ± d i α ∨ i exp   ±  q − q − 1  X n ≥ 1 h i, ± n z ± n   . and the formal delta function δ ( z ) = P n ∈ Z z n . When C is of ﬁnite type, the ab ove U q ( ˆ g ) is ca lled an ( u ntwiste d ) quant u m aﬃne algebr a ( without c entr al elements ) or quantu m lo op algebr a ; it is iso morphic to a sub quotient of the previo usly intro duced one (4.4) by the ideal gener a ted by c ± 1 / 2 − 1 [63, 64]. When C is of a ﬃne type, the qua ntum Kac- Mo o dy algebr a U q ( g ) is the o ne in (4.1). Its quantum a ﬃnization U q ( ˆ g ) is called a qu antum t or oidal algebr a (without central elements). In g e neral, if C is not of ﬁnite type, U q ( ˆ g ) is no longer isomorphic to a sub quotient of any quantum Ka c–Mo o dy algebra and has no Hopf algebra structure. F rom now on we sha ll exclusively consider a symmetriza ble ge ne r alized Car tan matrix C sa tisfying the follo wing condition due to Hernandez [8]: If C ij < − 1, then d i = − C j i = 1, (4.30) where D = diag ( d 1 , . . . , d r ) is the diagonal matrix symmetrizing C . W e say that a generalized Car tan matrix C is tamely lac e d if it is s ymmetrizable a nd satisﬁes the condition (4.30). A ge ne r alized Cartan matr ix C is simply lac e d if C ij = 0 or − 1 for any i 6 = j . If C is simply la c ed, then it is symmetric, d a = 1 for any a ∈ I , and it is tamely laced. With a tamely la ced generalized Cartan ma trix C , w e asso ciate a D ynkin diagr am in the standard wa y [1 1]: F or a ny pair i 6 = j ∈ I with C ij < 0, the vertices i and j are connected by max {| C ij | , | C j i |} lines, and the lines a re equipp ed with an ar r ow from j to i if C ij < − 1 . Note that the condition (4.30) means (i) the vertices i and j a re not connected if d i , d j > 1 and d i 6 = d j , (ii) the vertices i and j are connected by d i lines with an ar row from i to j o r not connected if d i > 1 and d j = 1, 43 (iii) the vertices i a nd j are connected by a s ing le line or no t co nnected if d i = d j . Example 4.10. (1) An y Cartan matrix of ﬁnite or aﬃne t y pe is tamely laced except for t y p es A (1) 1 and A (2) 2 ℓ . (2) The following gene r alized Cartan matrix C is tamely laced: C =     2 − 1 0 0 − 3 2 − 2 − 2 0 − 1 2 − 1 0 − 1 − 1 2     , D =     3 0 0 0 0 1 0 0 0 0 2 0 0 0 0 2     . The corresp onding Dynkin diagr a m is 1 2 3 4 Deﬁne the integer t b y t = lcm( d 1 , . . . , d r ) . F or a, b ∈ I , we wr ite a ∼ b if C ab < 0, i.e. a and b are adjacent in the corres p o nd- ing Dynkin diagr am. Let U b e either 1 t Z , the complex plane C , or the cylinder C ξ := C / (2 π √ − 1 /ξ ) Z for some ξ ∈ C \ 2 π √ − 1 Q , depending o n the situation under consideratio n. F or a tamely laced gener alized Carta n matrix C , the unre s tricted T-system asso ciated with C [8] is the following r elations among the commuting v ar ia bles { T ( a ) m ( u ) | a ∈ I , m ∈ Z ≥ 1 , u ∈ U } : T ( a ) m  u − d a t  T ( a ) m  u + d a t  = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) + Y b : b ∼ a T ( b ) d a d b m ( u ) if d a > 1 , (4.31) T ( a ) m  u − d a t  T ( a ) m  u + d a t  = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) + Y b : b ∼ a S ( b ) m ( u ) if d a = 1 , (4.32) where T ( a ) 0 ( u ) = 1 if they o ccur on the RHS in the rela tions. The symbol S ( b ) m ( u ) is deﬁned b y S ( b ) m ( u ) = d b Y k =1 T ( b ) 1+ E h m − k d b i  u + 1 t  2 k − 1 − m + E  m − k d b  d b  , (4.33) and E [ x ] ( x ∈ Q ) denotes the largest integer not ex ceeding x . Explicitly , S ( b ) m ( u ) is written as follows: F or 0 ≤ j < d b , S ( b ) d b m + j ( u ) = ( j Y k =1 T ( b ) m +1  u + 1 t ( j + 1 − 2 k )  )( d b − j Y k =1 T ( b ) m  u + 1 t ( d b − j + 1 − 2 k )  ) . F or example, for d b = 1, S ( b ) m ( u ) = T ( b ) m ( u ) , 44 for d b = 2, S ( b ) 2 m ( u ) = T ( b ) m  u − 1 t  T ( b ) m  u + 1 t  , S ( b ) 2 m +1 ( u ) = T ( b ) m +1 ( u ) T ( b ) m ( u ) , for d b = 3, S ( b ) 3 m ( u ) = T ( b ) m  u − 2 t  T ( b ) m ( u ) T ( b ) m  u + 2 t  , S ( b ) 3 m +1 ( u ) = T ( b ) m +1 ( u ) T ( b ) m  u − 1 t  T ( b ) m  u + 1 t  , S ( b ) 3 m +2 ( u ) = T ( b ) m +1  u − 1 t  T ( b ) m +1  u + 1 t  T ( b ) m ( u ) , and so o n. The second ter ms on the RHS of (4.31) a nd (4.32) ca n be written in a uniﬁed wa y as follows [8]: Y b : b ∼ a − C ab Y k =1 T ( b ) − C ba + E h d a ( m − k ) d b i  u + d b t  − 2 k + 1 C ab − C ba + E  d a ( m − k ) d b  − 1  − d a m t  . When C is of ﬁnite type g , the a b ove T-sy stem coinc ide s with the one for U q ( ˆ g ) in Section 2 .1. F or C of aﬃne t yp e , it was also s tudied by [79] as a discrete T o da ﬁeld equation. Let us pr o ceed to the Y-system. F or a tamely laced genera lized Ca r tan matrix C , the unrestricted Y-system asso ciated with C is the following relations among the commuting v ariables { Y ( a ) m ( u ) | a ∈ I , m ∈ Z ≥ 1 , u ∈ U } , wher e Y ( a ) 0 ( u ) − 1 = 0 if they o ccur on the RHS in the rela tio ns: Y ( a ) m  u − d a t  Y ( a ) m  u + d a t  = Q b : b ∼ a Z ( b ) d a d b ,m ( u ) (1 + Y ( a ) m − 1 ( u ) − 1 )(1 + Y ( a ) m +1 ( u ) − 1 ) if d a > 1 , (4.34) Y ( a ) m  u − d a t  Y ( a ) m  u + d a t  = Q b : b ∼ a  1 + Y ( b ) m d b ( u )  (1 + Y ( a ) m − 1 ( u ) − 1 )(1 + Y ( a ) m +1 ( u ) − 1 ) if d a = 1 , (4.35) where for p ∈ Z ≥ 1 Z ( b ) p,m ( u ) = p − 1 Y j = − p +1    p −| j | Y k =1  1 + Y ( b ) pm + j  u + 1 t ( p − | j | + 1 − 2 k )     , and Y ( b ) m d b ( u ) = 0 in (4.35) if m d b 6∈ Z ≥ 1 . The Y-systems here ar e formally in the same for m as (2.1 1)–(2.15) for the q uan- tum aﬃne alg ebras. How ever, p in Z ( b ) p,m ( u ) here may be greater than 3. On the RHS of (4.3 4), d a d b is e ither 1 or d a due to (4.3 0). The term Z ( b ) p,m ( u ) is wr itten more explicitly as follows: for p = 1 , Z ( b ) 1 ,m ( u ) = 1 + Y ( b ) m ( u ) , 45 for p = 2, Z ( b ) 2 ,m ( u ) =  1 + Y ( b ) 2 m − 1 ( u )   1 + Y ( b ) 2 m  u − 1 t   1 + Y ( b ) 2 m  u + 1 t   1 + Y ( b ) 2 m +1 ( u )  , for p = 3, Z ( b ) 3 ,m ( u ) =  1 + Y ( b ) 3 m − 2 ( u )   1 + Y ( b ) 3 m − 1  u − 1 t   1 + Y ( b ) 3 m − 1  u + 1 t  ×  1 + Y ( b ) 3 m  u − 2 t   1 + Y ( b ) 3 m ( u )   1 + Y ( b ) 3 m  u + 2 t  ×  1 + Y ( b ) 3 m +1  u − 1 t   1 + Y ( b ) 3 m +1  u + 1 t   1 + Y ( b ) 3 m +2 ( u )  , and so on. There ar e p 2 factors in Z ( b ) p,m ( u ). The T a nd Y- s ystems in this s ubsection satisfy fo rmally the sa me relations as those e xplained in Section 2.3. Their restricted v ersions ha ve a ls o b een formulated in [78]. 4.7. Bibliog raphical notes. The orig in o f the Kirillov-Reshetikhin mo dules (they are named so in [80, Deﬁnition 1.1 ]) go es back to [81], where the s pe c tral parameter depe ndence was not co nsidered. The idea of treating them as one family of Y ( g ) or U q ( ˆ g ) mo dules with sp ectral pa rameter sa tisfying the T-system in the Grothendieck ring was initiated by [1], where the identiﬁcation by Drinfeld p olynomials was also given in the con text of Y angian based o n the result of [6 9]. Mea nwhile, the representation theor y of ﬁnite dimensiona l U q ( ˆ g ) mo dules was pushed forward by [82, 65], where the Kir illov-Reshetikhin mo dules were characterized and studied as minimal aﬃnizations of U q ( ˆ g ) mo dules [83, 84, 85, 86, 87]. The rela tio n betw e e n the Kir illov-Reshetikhin mo dules a nd T-systems b eca me transpare nt after the intro duction of q -character by [70]. The case of Y angian go es back to [71]. Theorem 4.8 is due to [67] for simply laced g a nd [68] for general g . Under certain cir cumstances, there ar e algor ithms to compute q -characters [72] or its further gener alization called t -ana log of q - character s χ q,t [88, 89] for ADE case. In particular, χ q,t of all the fund amental repre sentations has b een pro duced [89], among whic h the E 8 case requires a sup ercomputer. The T-systems for the q ua nt um aﬃnizations of quantum Kac–Mo o dy a lgebras in Section 4 .6 are due to [8]. The corr e sp o nding Y-system and formulation by cluster algebra are given in [78]. 5. Formula tion by cluster algebras 5.1. Dilogarithm ide n titi es i n conformal ﬁe l d theory. Let L ( x ) be the R o gers dilo garithm function [90, 91] L ( x ) = − 1 2 Z x 0  ln(1 − y ) y + ln y 1 − y  dy (0 ≤ x ≤ 1) . (5.1) 46 It is w ell known that the following prop erties hold (0 ≤ x, y ≤ 1). L (0) = 0 , L (1) = π 2 6 , (5.2) L ( x ) + L (1 − x ) = π 2 6 , (5.3) L ( x ) + L ( y ) + L (1 − xy ) + L  1 − x 1 − xy  + L  1 − y 1 − xy  = π 2 2 . (5.4) In the series of works by Ba zhanov, Kir illov, a nd Reshetikhin [5 3, 37, 81, 92, 59], they reached a remar k able conjecture o n ident ities expressing the c ent ral ch arges of conformal ﬁeld theories in terms of L ( x ), and pa rtly established it. In wha t follows, g denotes any one of the simple Lie algebra s A r , B r , . . . , G 2 as in the pr e vious sections. In Section 2 .2, we deﬁned the level ℓ restricted Y- system for g for ℓ ∈ Z ≥ 1 . Let us intro duce the system of relations for the v ariable { Y ( a ) m | a ∈ I , 1 ≤ m ≤ t a ℓ − 1 } o btained fro m the level ℓ restricted Y-system by setting Y ( a ) m ( u ) = Y ( a ) m dropping the dep endence o n the sp ectral pa rameter u . W e call it the level ℓ r estricte d c onstant Y-syst em . Theorem 5. 1 ([9 3, 94]) . Ther e exists a u nique solution of the level ℓ r estricte d c onstant Y-system for g satisfying Y ( a ) m ∈ R > 0 for al l a ∈ I , 1 ≤ m ≤ t a ℓ − 1 . Theorem 5.1 was pr ov ed by [9 3] for simply laced case , and extended to nonsimply laced case by [94] using the sa me metho d. F or mor e information on the constant Y-system, see Section 14.4 and 14.6. The following theorem was origina lly conjectured by [81] and [59] for s imply laced ca s e, a nd co njectured by [92] a nd pr op erly corr ected by [18] fo r nonsimply laced case. Theorem 5.2 (Dilog arithm identities [92, 9 5, 94, 96 ]) . Supp ose that a fa mily of p ositive r e al nu mb ers { Y ( a ) m | a ∈ I , 1 ≤ m ≤ t a ℓ − 1 } satisfy the leve l ℓ c onstant Y-system for g . Then, t he fol lowing identities hold: 6 π 2 X a ∈ I t a ℓ − 1 X m =1 L Y ( a ) m 1 + Y ( a ) m ! = ℓ dim g ℓ + h ∨ − ra nk g , (5.5) wher e h ∨ is the du al Coxeter numb er of g (2.3). The ra tional num b er of the ﬁr s t term o n the RHS of (5.5) is the central c ha rge of the Wess- Zumino-Witten c onformal ﬁeld the ory a sso ciated with g with level ℓ [97, 98]. The ra tional num b er o n the RHS of (5.5 ) itself is also the central charge of the p ar afermion c onformal ﬁeld t he ory asso ciated with g with level ℓ [99, 100]. The ident it y (5.5) is crucial to e stablish the co nnection betw e en co nformal ﬁeld theor ie s and v ar ious types of non confor mal integrable mo dels in v arious limits (cf. Section 15.3). Example 5.3 ([53]) . Cons ider the c a se g = A 1 and any ℓ , whic h is equiv alent to the case g = A ℓ − 1 and ℓ = 2 by the level-r ank duality . Then, one has the solution Y (1) m = sin 2 π ℓ +2 sin mπ ℓ +2 sin ( m +2) π ℓ +2 , (5.6) 47 and the corresp o nding identit y (5.5) reads 6 π 2 ℓ − 1 X m =1 L sin 2 π ℓ +2 sin 2 ( m +1) π ℓ +2 ! = 3 ℓ 2 + ℓ − 1 . (5.7) This iden tity has b een known and s tudied by v arious a uthors in v ar ious p oints of view. See [1 0 1, 102] a nd reference ther ein. In particular , the identit y is der ived [103, 104] from the following q -ser ies expr ession [105] for the parafermio n conforma l character (“string function” in [11] multiplied with Dedekind’s eta function): ∞ X n 1 ,...,n ℓ − 1 =0 q P ℓ − 1 k,m =1 n k n m (min( k,m ) − km ℓ ) ℓ − 1 Y m =1 ( q ) − 1 n m , ( q ) k := k Y j =1 (1 − q j ) , (5.8) where the sum is under the constr aint P ℓ − 1 m =1 mn m ≡ 0 mo d 2 ℓ . In fact, a crude estimate by a saddle p oint method tells that a s q → 1 , this series diverges as const · ( q ) − c/ 24 where c is the LHS of (5.7) a nd q → 0 is the modular conjugate sp eciﬁed by (ln q )(ln q ) = 4 π 2 . Co mparing this with the known asymptotics of the string function [11] yields (5.7). F or general g , see around (14.43). F or g = A r , Kirillov [92] gav e the e x plicit expression of the so lutio n (cf. Example 14.4), and prov ed the cor resp onding ident it y (5.5) by the a nalytic metho d, but an extension of the pro of to the other case s se e med diﬃcult. In the 19 90s, p e ople pursued a pr o of through lifting the dilogarithm identities to the Ro gers-Ra ma nuj a n type identities as Ex ample 5.3 (e.g. [106, 10 7, 108, 109, 110]). This crea ted a new sub ject called the F ermionic formula of conformal char- acters and their v ariants, which turned out to b e a r ich sub ject itself, a nd it has bee n intensively s tudied to this day by its own r ight. See (ii) in Section 13.8. In spite of this succe s sful developmen t, the o riginal pr oblem of pr oving the dilogarithm ident ities (5.5) itself did not make muc h pr ogres s. The s c ene changed after the introduction of a new class of commutativ e algebra s called cluster algebr as by F omin-Z e le vinsky [111] ar ound 2 000, which we ex plain in this section. 5.2. Cluster alg ebras wi th co eﬃci e n ts. Here we recall the deﬁnition of the cluster alge br as with co eﬃcients and some of their bas ic prop erties, following the conv ention in [9] with slight change of no tations and terminology . See [9 ] for more detail and information. Fix a n arbitra ry semiﬁeld P , i.e. an ab elia n multiplicativ e gro up e ndowed with a binary op er ation of addition ⊕ which is commutativ e, asso ciative, a nd distributiv e with r e sp e c t to the m ultiplicatio n [112]. Let QP denote the quotient ﬁeld of the group ring ZP of P . Let I b e a ﬁnite set 19 , and let B = ( B ij ) i,j ∈ I be a skew symmetrizable (integer) matrix; namely , there is a diag onal p ositive integer matrix D such that t ( D B ) = − D B . Let x = ( x i ) i ∈ I be an I -tuple o f for mal v ariables, a nd let y = ( y i ) i ∈ I be an I -tuple of elements in P . F or the triplet ( B , x, y ), called the initial se e d , the clust er algebr a A ( B , x, y ) with c o eﬃcients in P is deﬁned as fo llows. Let ( B ′ , x ′ , y ′ ) be a tr iplet consisting of skew s ymmetrizable matrix B ′ , an I - tuple x ′ = ( x ′ i ) i ∈ I with x ′ i ∈ QP ( x ), and an I -tuple y ′ = ( y ′ i ) i ∈ I with y ′ i ∈ P . 19 This I does not necessarily corr espond to the I in Section 2.1 for the index set of Dynkin diagrams. 48 F or eac h k ∈ I , w e deﬁne ano ther triplet ( B ′′ , x ′′ , y ′′ ) = µ k ( B ′ , x ′ , y ′ ), called the mutation of ( B ′ , x ′ , y ′ ) at k , as follows. (i) Mutations of the matrix. B ′′ ij = ( − B ′ ij i = k or j = k , B ′ ij + 1 2 ( | B ′ ik | B ′ kj + B ′ ik | B ′ kj | ) otherwise . (5.9) (ii) Exchange r elation of the c o eﬃcient tuple. y ′′ i =          y ′ k − 1 i = k , y ′ i 1 (1 ⊕ y ′ k − 1 ) B ′ ki i 6 = k , B ′ ki ≥ 0 , y ′ i (1 ⊕ y ′ k ) − B ′ ki i 6 = k , B ′ ki ≤ 0 . (5.10) (iii) Exchange r elation of the cluster. x ′′ i =      y ′ k Q j : B ′ jk > 0 x ′ j B ′ jk + Q j : B ′ jk < 0 x ′ j − B ′ jk (1 ⊕ y ′ k ) x ′ k i = k . x ′ i i 6 = k , (5.11) It is easy to see that µ k is an inv olution, namely , µ k ( B ′′ , x ′′ , y ′′ ) = ( B ′ , x ′ , y ′ ). Now, starting fro m the initial se e d ( B , x, y ), iter ate mutations and collect all the resulted triplets ( B ′ , x ′ , y ′ ). W e call ( B ′ , x ′ , y ′ ) the se e ds , y ′ and y ′ i a c o eﬃcient tuple a nd a c o eﬃcient , x ′ and x ′ i , a cluster a nd a cluster variab le , r esp ectively . The cluster algebr a A ( B , x, y ) with c o eﬃcients in P is the ZP -subalgebra of the r ational function ﬁeld QP ( x ) genera ted by all the cluster v ariables. It is standard to iden tify a skew-symmetric (integer) ma trix B = ( B ij ) i,j ∈ I with a quiver Q without lo ops or 2-cycles . The s et of the vertices of Q is given by I , and we put B ij arrows from i to j if B ij > 0. The mutation Q ′′ = µ k ( Q ′ ) of a quiver Q ′ is g iven by the following r ule: F or each pair of an incoming arrow i → k and an outgoing arrow k → j in Q ′ , add a new arrow i → j . Then, remove a maximal set of pairwise disjoint 2-cyc les. Finally , reverse all a rrows incident with k . Let P univ ( y ) b e the u niversal semiﬁeld of the I -tuple o f g enerator s y = ( y i ) i ∈ I , namely , the semiﬁeld consisting of the subtr action-fr e e rationa l functions of formal v ariables y with usual mu ltiplication and addition in the rational function ﬁeld Q ( y ). W e wr ite ⊕ in P univ ( y ) as + for simplicity . F r om n ow on, un less otherwise mentione d, we set the semiﬁeld P for A ( B , x, y ) to b e P univ ( y ) , wher e y is the c o eﬃcient tuple in the initial se e d ( B , x, y ) . Let P trop ( y ) b e the tr opic al semiﬁeld of y = ( y i ) i ∈ I , which is the ab elian multi- plicative gr oup free ly generated by y endowed with the addition ⊕ Y i y a i i ⊕ Y i y b i i = Y i y min( a i ,b i ) i . (5.1 2) There is a cano nical surjective semiﬁeld homomor phism π T (the tr opic al evaluation ) from P univ ( y ) to P trop ( y ) deﬁned by π T ( y ) = y . F or any co eﬃcient y ′ i of A ( B , x, y ), let us write [ y ′ i ] T := π T ( y ′ i ) fo r simplicity . W e call [ y ′ i ] T ’s the tr opic al c o eﬃcients (the princip al c o eﬃcients in [9]). They satisfy the exchange rela tion (5.10) by replacing y ′ i with [ y ′ i ] T with ⊕ b eing the addition in (5.12). W e also extend this ho - momorphism to the homomor phis m of ﬁelds π T : ( QP univ ( y ))( x ) → ( Q P trop ( y ))( x ). T o ea ch seed ( B ′ , x ′ , y ′ ) of A ( B , x, y ) we attach the F -p olynomials F ′ i ( y ) ∈ Q ( y ) ( i ∈ I ) by the sp ecializatio n of [ x ′ i ] T at x j = 1 ( j ∈ I ). It is, in fact, a p olyno mial 49 in y with integer co eﬃcients due to the Laurent phenomenon [9, Prop o s ition 3.6]. F or deﬁniteness, let us ta ke I = { 1 , . . . , n } . Then, x ′ and y ′ hav e the following factorized expressions [9, Pr o p osition 3.1 3, Corollar y 6.3] by the F -p olyno mials. x ′ i =   n Y j =1 x g ′ ji j   F ′ i ( ˆ y 1 , . . . , ˆ y n ) F ′ i ( y 1 , . . . , y n ) , ˆ y i = y i n Y j =1 x B ji j , (5.13) y ′ i = [ y ′ i ] T n Y j =1 F ′ j ( y 1 , . . . , y n ) B ′ ji . (5.14) The integer vector g ′ i = ( g ′ 1 i , . . . , g ′ ni ) ( i = 1 , . . . , n ) uniquely determined by (5.13) for each x ′ i is called the g - ve ctor for x ′ i . Let i = ( i 1 , . . . , i r ) be an I -sequence, namely , i 1 , . . . , i r ∈ I . W e deﬁne the c om- p osite mut ation µ i by µ i = µ i r · · · µ i 2 µ i 1 , where the pr o duct means the comp osition. Lemma 5.4 . L et B = ( B ij ) i,j ∈ I b e a skew symmetrizable matrix and let i = ( i 1 , . . . , i r ) b e an I -se quenc e. Supp ose that B i a i b = 0 f or any 1 ≤ a, b ≤ r . Then, the fol lowing facts hold. (a) F or any p ermutat ion σ o f { 1 , . . . , r } , we have µ i ( B , x, y ) = µ ( i σ (1) ,...,i σ ( r ) ) ( B , x, y ) . (5.15) (b) L et B ′ = µ i ( B ) . Then, B ′ i a i b = 0 holds for any 1 ≤ a, b ≤ r . (c) L et ( B ′ , x ′ , y ′ ) = µ i ( B , x, y ) . Then, ( B , x, y ) = µ i ( B ′ , x ′ , y ′ ) . 5.3. T and Y-systems i n cluster algebras. All the T and Y-systems in Sec- tions 2.1 – 2.5 are rega r ded as rela tions among a cluster among cluster v ar iables and co eﬃcients in cer ta in cluster algebras A ( B , x, y ). Let us mention tw o big adv antages of cluster algebra formulation. (a) The T a nd Y-systems a r e integrated in one alg ebra A ( B , x, y ), and com- monly c o ntrolled by F - p o lynomials (tog ether with tropical co eﬃcients and g -vectors) thro ugh the formulas (5.13) and (5.14). This fact may b e har dly realized just by trea ting the T and Y-systems only . (b) The cluster algebra A ( B , x, y ) itself is further controlled by the (gener alize d) cluster c ate gory developed by [11 3, 114, 1 1 5, 116, 1 17, 118]. Here w e concentrate on a n example of level 4 restricted T and Y-systems for A 4 to present a basic idea. Let Q b e the following quiver with index set I = { 1 , 2 , 3 , 4 } × { 1 , 2 , 3 } . Note that we als o attached the pr op erty + / − to each vertex. ✻ ❄ ❄ ✻ ✻ ❄ ❄ ✻ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ + − + − − + − + + − + − (1 , 1) (2 , 1) (3 , 1) (4 , 1) (1 , 2) (2 , 2) (3 , 2) (4 , 2) (1 , 3) (2 , 3) (3 , 3) (4 , 3) (5.16) Below we ident ify Q with the cor resp onding skew symmetric matrix B as describ ed in Section 5.2. Let i + (resp. i − ) b e a sequence o f all the distinct ele men ts of I with pro per ty + (resp. − ), where the order of the s equence is chosen arbitr a rily tha nk to Le mma 5.4. 50 Then, the quiver Q has the following per io dicity under the sequences of mutation i + and i − : Q µ i + ← → Q op µ i − ← → Q, (5.17) where Q op is the opp osite quiver of Q , namely , the quiver obtained from Q by inv erting all the arrows. Now we set ( Q (0) , x (0) , y (0)) := ( Q, x, y ) (the initial seed of A ( Q, x, y )) a nd consider the corres p o nding inﬁnite sequence of mutations of se e ds · · · µ i + ← → ( Q ( − 1) , x ( − 1) ,y ( − 1 )) µ i − ← → ( Q (0) , x (0) , y (0)) µ i + ← → ( Q (1) , x (1) , y (1)) µ i − ← → ( Q (2) , x (2) , y (2)) µ i + ← → · · · , (5.18) Q ( u ) = ( Q u is even Q op u is o dd , (5.19) thereby intro ducing a family of clusters x ( u ) ( u ∈ Z ) and co eﬃcients tuples y ( u ) ( u ∈ Z ). F or (( i, i ′ ) , u ) ∈ I × Z , we write (( i, i ′ ) , u ) : p + if i + i ′ + u is e ven, or equiv alently , if u is even and ( i, i ′ ) has the prop erty +, o r u is o dd and ( i, i ′ ) has the prop erty − . Plainly sp eaking , (( i, i ′ ) , u ) : p + is a forwar d mut ation p oint in (5.18). F or (( i, i ′ ) , u ) ∈ I × Z , w e set (( i, i ′ ) , u ) : ˜ p + if (( i, i ′ ) , u + 1) : p + . Consequently , we hav e (( i, i ′ ) , u ) : ˜ p + ⇐ ⇒ (( i, i ′ ) , u ± 1) : p + . (5.20) First, we explain ho w the Y-system app ear s in cluster a lgebra. The sequence of mutations (5.18) gives v arious relations a mo ng co eﬃcient s y i,i ′ ( u ) ((( i, i ′ ) , u ) ∈ I × Z ) by the exchange r e lation (5 .10). Then, all these co eﬃcients a r e pr o ducts of the “genera ting” co e ﬃcie n ts y i,i ′ ( u ) and 1 + y i,i ′ ( u ) ((( i, i ′ ) , u ) : p + ). F urthermore , these generating co eﬃcients o be y some relations, whic h are the Y-system. Let us write down the r elations explicitly . T ak e (( i, i ′ ) , u ) : p + and consider the m utation at (( i, i ′ ) , u ), where y i,i ′ ( u ) is e x changed to y i,i ′ ( u + 1 ) = y i,i ′ ( u ) − 1 , b y (5.10). In the next step going from Q ( u + 1) to Q ( u + 2), the (forward) m utation po int s are those sa tis fying (( j, j ′ ) , u + 1 ) : p + . Therefor e the ab ov e y i,i ′ ( u + 1) g ets m ultiplied by factors (1 + y j,j ′ ( u + 1)) if the quiver Q ( u + 1) ha s an arrow from ( i, i ′ ) to ( j, j ′ ), and (1 + y j,j ′ ( u + 1) − 1 ) − 1 if the quiver Q ( u + 1) has a n a r row from ( j, j ′ ) to ( i, i ′ ). The r esult coincides with the co eﬃcient y i,i ′ ( u + 2). In summar y , we hav e the following r elations: F or (( i, i ′ ) , u ) : p + , y i,i ′ ( u ) y i,i ′ ( u + 2) = (1 + y i − 1 ,i ′ ( u + 1))(1 + y i +1 ,i ′ ( u + 1)) (1 + y i,i ′ − 1 ( u + 1) − 1 )(1 + y i,i ′ +1 ( u + 1) − 1 ) , (5.21) where y 0 ,i ′ ( u + 1) = y 5 ,i ′ ( u + 1) = 0 and y i, 0 ( u + 1) − 1 = y i, 4 ( u + 1) − 1 = 0 o n the RHS. Or, equiv alently , for (( i, i ′ ) , u ) : ˜ p + , y i,i ′ ( u − 1) y i,i ′ ( u + 1) = (1 + y i − 1 ,i ′ ( u ))(1 + y i +1 ,i ′ ( u )) (1 + y i,i ′ − 1 ( u ) − 1 )(1 + y i,i ′ +1 ( u ) − 1 ) . (5 .22) This certainly agrees with the level 4 restricted Y-s y stem for A 4 under the identi- ﬁcation of y i,i ′ ( u ) with Y ( i ) i ′ ( u ). Next, we explain how the T-sys tem appear s in cluster algebra. The sequence of m utations (5.18) gives v ario us relations among cluster v aria bles x i,i ′ ( u ) ((( i, i ′ ) , u ) ∈ I × Z ) b y the exchange relation (5.11). All these co e ﬃcie n ts a re r e presented by 51 the “genera ting” cluster v ar iables x i,i ′ ( u ) ((( i, i ′ ) , u ) : p + ). F urthermore, these generating cluster v ariables ob ey some rela tions, which are the T-system. Let us write down the rela tions explicitly . T a ke (( i, i ′ ) , u ) : p + and cons ide r the m utation at (( i, i ′ ) , u ). Then, by (5.11) and the fact that (( i ± 1 , i ′ ) , u ) and (( i, i ′ ± 1) , u ) are not forward mutation points, we hav e x i,i ′ ( u ) x i,i ′ ( u + 2) = y i,i ′ ( u ) 1 + y i,i ′ ( u ) x i − 1 ,i ′ ( u + 1) x i +1 ,i ′ ( u + 1) + 1 1 + y i,i ′ ( u ) x i,i ′ − 1 ( u + 1) x i,i ′ +1 ( u + 1) , (5.23) where x 0 ,i ′ ( u + 1) = x 5 ,i ′ ( u + 1) = x i, 0 ( u + 1) = x i, 4 ( u + 1) = 1 on the RHS. By int ro ducing the “shifted cluster v ar ia bles” ˜ x i ( u ) := x i ( u + 1) for (( i, i ′ ) , u ) : ˜ p + , these relations ca n be wr itten in a mo re “ balanced” for m and b ecome pa r allel to (5.22) as follows: F or (( i, i ′ ) , u ) : p + , ˜ x i,i ′ ( u − 1) ˜ x i,i ′ ( u + 1) = y i,i ′ ( u ) 1 + y i,i ′ ( u ) ˜ x i − 1 ,i ′ ( u ) ˜ x i +1 ,i ′ ( u ) + 1 1 + y i,i ′ ( u ) ˜ x i,i ′ − 1 ( u ) ˜ x i,i ′ +1 ( u ) . (5.24) Let A ( B , x ) be the cluster a lgebra with trivial c o eﬃcients with initial s e e d ( B , x ). Namely , we set every co eﬃcient to b e 1 in the tr ivial s emiﬁeld 1 = { 1 } . Let π 1 : P univ ( y ) → 1 b e the pr o jection. Let [ x i ( u )] 1 be the ima ge o f x i ( u ) by the algebr a homomorphism A ( B , x, y ) → A ( B , x ) induced from π 1 . By the sp ecializ ation of (5.24), we hav e [ ˜ x i,i ′ ( u − 1)] 1 [ ˜ x i,i ′ ( u + 1)] 1 = [ ˜ x i − 1 ,i ′ ( u )] 1 [ ˜ x i +1 ,i ′ ( u )] 1 + [ ˜ x i,i ′ − 1 ( u )] 1 [ ˜ x i,i ′ +1 ( u )] 1 . (5.25) This certainly agr ees with the level 4 restr icted T-system for A 4 under the identi- ﬁcation of [ ˜ x i,i ′ ( u )] 1 with T ( i ) i ′ ( u ). F or g simply laced, the quiver rele v a n t to the level ℓ restr icted T a nd Y-sy s tems is drawn similar ly to (5.16) on the vertex set I = { no des of the Dynkin diagr am } × { 1 , 2 , . . . , ℓ − 1 } . F or g nonsimply la ced, it is slightly mo re in volv ed [94, 96]. Here we only give e x amples for B 3 with level 2 (left) and level 3 (right). ✛ ✛ ✲ ✛ ❄ ✻ ❥ ✯ ✲ ✛ ✲ ✲ ✛ ✛ ✲ ✛ ❄ ✻ ❄ ✻ ✻ ❄ ✻ ❄ ✙ ❨ ❥ ✯ Remark 5. 5. Once we realize that the T a nd Y-systems a re integrated in a single cluster alg ebra with co e ﬃcient s as a bove, the r elation b etw een T and Y-sys tems in Theorem 2.5 be c omes a n immediate consequence o f a more general rela tio n b etw een cluster v ariables a nd co eﬃcients in [9, Pro p. 3.9], wher e (2.1 9) is a sp ecial ca se o f [9, eq. (3.7)] with the sp ecializ ation of the base semiﬁeld P therein to the trivial semiﬁeld. See als o [1 19, Prop. 5 .11] for the r e la tion b etw ee n more general T and Y-systems. 52 5.4. Application to p erio dicit y and dil ogarithm identities. As re mark able applications of the clus ter a lgebra formulation, one can prov e the p erio dicities of T and Y-systems and dilogar ithm identities (5.5). The following pe rio dicity pro p e rty was orig inally conjectured for type A 1 by [3], for simply laced case by Rav anini-T ateo- V alleriani [5], and fo r nonsimply laced case by Kuniba-Nak anis hi-Suzuki [1 ]. Theorem 5.6 (Perio dicity [120, 121, 1 22, 123, 12 4, 115, 12 5, 1 7, 94, 96]) . F or any family of variables { Y ( a ) m ( u ) | a ∈ I , 1 ≤ m ≤ t a ℓ − 1 , u ∈ Z } satisfying the level ℓ r estricte d Y - system for g , one has the p erio dicity Y ( a ) m ( u + 2( h ∨ + ℓ )) = Y ( a ) m ( u ) . (5.26) T o prov e Theorem 5.6 in full gener ality , the use o f the categoriﬁcation of the cluster algebra by the cluster ca teg ory b y [117, 118] is essential. Since the T-system is integrated in the sa me cluster algebra, o ne can simul- taneously prove the perio dicity o f T-system a s well, which was ov erlo o ked in the literature un til r ecently [126, 1 7]. Theorem 5. 7 (Perio dicity [9, 127, 124, 115, 17, 94, 96]) . F or any family of variables { T ( a ) m ( u ) | a ∈ I , 1 ≤ m ≤ t a ℓ − 1 , u ∈ Z } satisfying the level ℓ r est ricte d T-system for g , one has t he p erio dicity T ( a ) m ( u + 2( h ∨ + ℓ )) = T ( a ) m ( u ) . (5.27 ) Closely r e la ted to the p erio dicity of Y-systems, the following (signiﬁca nt ) func- tional g eneralizatio n of the dilo garithm identities (5.5 ) was originally c onjectured for simply laced case by Glio z z i-T ateo [128]. Theorem 5.8 (F unctional diloga rithm identities [1 20, 121, 129, 95, 94, 96]) . Su p- p ose that a family of p ositive r e al nu mb ers { Y ( a ) m ( u ) | a ∈ I , 1 ≤ m ≤ t a ℓ − 1 , u ∈ Z } satisfy the level ℓ r est ricte d Y -system for g . Then, t he fol lowing identities hold: 6 π 2 X a ∈ I t a ℓ − 1 X m =1 2( h ∨ + ℓ ) − 1 X u =0 L Y ( a ) m ( u ) 1 + Y ( a ) m ( u ) ! = 2( ℓh − h ∨ )rank g , (5.28) wher e h is t he Coxeter n u mb er of g (2.3). Example 5.9 ([128]) . (i) In the simples t case, type A 1 , the identit y (5.28) is equiv alent to (5.3). (ii) In the nex t simplest case, type A 2 , the iden tity (5.28) is equiv ale nt to the 5-term relation (5.4). Theorem 5.8 implies Theor em 5.2; namely , take a c onstant solution Y ( a ) m = Y ( a ) m ( u ) of the Y-s y stem with resp ect to the sp ectr al para meter u . O r equiv a- lent ly , tak e a solution to the constant Y-system in Section 1 4.4. Then, one obta ins (5.5) from (5.28). See Section 5 .5 fo r more precise acco unt of contributions to Theor ems 5 .6, 5.7, and 5.8. 5.5. Bibliog raphical no te s . The cluster algebra ic formulation of Y-systems was given for the simply laced case with level 2 by [1 22], for the simply laced case with ge neral level by [115], for the nonsimply laced case b y [94, 96], and for the quantum aﬃnizatio ns of the tamely lac e d quan tum Ka c-Mo o dy algebras b y [78, 53 130]. The re c ognition of T-systems in the clus ter algebra s was ma de a little later than Y-systems in [1 31, 17, 132], thoug h the simply laced case with level 2 clearly app eared in [13 3]. The formulation here is due to [78, 94, 96]. See [11 9] for a further generaliza tion of T and Y-sys tems in view of cluster algebras . Theorem 5.6 was proved for t y pe A r with le vel 2 by [120, 121], for the simply laced c a se with level 2 by [133], for type A r with gener al level by [123] and [124], for the simply laced case with general level by [1 15, 12 5], and for all the cases with uniﬁed metho d by [94, 96]. Theorem 5.7 was proved for the simply laced case with level 2 by [133], for type A r with ge ne r al level by [127] and [124], for the simply laced case with gener al level by [115] and [17], fo r t y pe C r with general level by [17], and for a ll the c a ses with a uniﬁed metho d b y [94, 9 6]. Actually in [17, 94, 96], reﬁnements of Theore m 5 .6 and 5.7 hav e b een obtained concerning the pr op erty under the half shift u → u + h ∨ + ℓ . Theorem 5.8 was proved for t y pe A r with le vel 2 by [120, 121], for the simply laced case with level 2 b y [129], for the simply laced case with g eneral level by [95] and for the nonsimply la ced case by [94, 96]. See [11 9] for a further genera liz ation of dilogarithm identit ies in view of cluster alg ebras. There is a dilogar ithm conjecture that generalize s (5.5) inv o lving − 24 × (scaling dimensions) in addition to the central c ha rge on the RHS. See [4] and [1 34, app endix D]. Some of them has b een prov e d in [10 1, section 1.3, 1.4]. 6. Jacobi-Tr udi type formula 6.1. In tro ductio n: T yp e A r . In this section we exclusively co nsider unr estricted T-systems. By Theorem 4.3, w e know that T ( a ) m ( u ) is ex pr essible as a p olynomial in the fundamental ones T (1) 1 ( v ) , . . . , T ( r ) 1 ( v ) with v arious v . Such form ula s can b e derived directly . Consider for instance the unrestric ted T-system for A 2 : T (1) m ( u − 1) T (1) m ( u + 1) = T (1) m − 1 ( u ) T (1) m +1 ( u ) + T (2) m ( u ) , T (2) m ( u − 1) T (2) m ( u + 1) = T (2) m − 1 ( u ) T (2) m +1 ( u ) + T (1) m ( u ) . Setting m = 1 , 2 and noting T (1) 0 ( u ) = T (2) 0 ( u ) = 1, o ne gets T (1) 2 ( u ) = T (1) 1 ( u − 1) T (1) 1 ( u + 1) − T (2) 1 ( u ) , T (2) 2 ( u ) = T (2) 1 ( u − 1) T (2) 1 ( u + 1) − T (1) 1 ( u ) , T (1) 3 ( u ) = T (1) 1 ( u − 2) T (1) 1 ( u ) T (1) 1 ( u + 2) − T (1) 1 ( u − 2) T (2) 1 ( u + 1) − T (1) 1 ( u + 2) T (2) 1 ( u − 1) + 1 . The formulas gener ated in this manner are systematized in a determina nt form: T (1) 2 ( u ) =      T (1) 1 ( u − 1) T (2) 1 ( u ) 1 T (1) 1 ( u + 1)      , T (2) 2 ( u ) =      T (2) 1 ( u − 1) 1 T (1) 1 ( u ) T (2) 1 ( u + 1)      , T (1) 3 ( u ) =        T (1) 1 ( u − 2) T (2) 1 ( u − 1) 1 1 T (1) 1 ( u ) T (2) 1 ( u + 1) 0 1 T (1) 1 ( u + 2)        . Pro ceeding similarly , one ge ts 54 Theorem 6. 1 ([59]) . F or t he u n r estricte d T-system for A r , the fol lowing formula is valid: T ( a ) m ( u ) = det( T ( a − i + j ) 1 ( u + i + j − m − 1)) 1 ≤ i,j ≤ m , (6.1) wher e T ( a ) 1 ( u ) = 0 u nless 0 ≤ a ≤ r + 1 , and T (0) 1 ( u ) = T ( r +1) 1 ( u ) = 1 . The pro of reduces to the Jaco bi iden tity a mong the determinants D [ m +1 m +1 ] D [ 1 1 ] = D [ 1 ,m +1 1 ,m +1 ] D + D [ 1 m +1 ] D [ m +1 1 ] , (6.2) where D [ i 1 ,i 2 ,... j 1 ,j 2 ,... ] is the minor of D removing i k ’s rows and j k ’s columns. Alternatively , one can also s olve the T-s ystem to express everything by T (1) k ( v ) with v ar ious v and k . By the sa me metho d a s b efore, o ne can ea sily sy stematize such formulas and establish Theorem 6.2 ([59]) . F or the unr estricte d T-s yst em for A r (2.5) without assuming T ( r +1) m ( u ) = 1 , the fol lowing formula is valid: T ( a ) m ( u ) = de t( T (1) m − i + j ( u + i + j − a − 1)) 1 ≤ i,j ≤ a (1 ≤ a ≤ r + 1 ) , (6.3) wher e T (1) 0 ( u ) = 1 and T (1) m ( u ) = 0 for m < 0 . The fo r mulas (6 .1) and (6.3) a re quantum analog o f the Jacobi-T r udi formula for Sch ur functions [135]. In the rema inder of this se ction, we pre s ent the Jacobi-T rudi type formulas analogo us to (6.1) for the T-systems for B r , C r and D r . The result in volves no t only deter minants but also Pfaﬃans for T ( r ) m ( u ) in C r and T ( r − 1) m ( u ) and T ( r ) m ( u ) in D r . 6.2. T yp e B r . F or any k ∈ C , se t x a k = ( T ( a ) 1 ( u + k ) 1 ≤ a ≤ r , 1 a = 0 . (6.4) W e introduce the inﬁnite dimensional matrices T = ( T ij ) i,j ∈ Z and E = ( E ij ) i,j ∈ Z as follows. T ij =              x j − i 2 +1 i + j 2 − 1 if i ∈ 2 Z + 1 and i − j 2 ∈ { 1 , 0 , . . . , 2 − r } , − x i − j 2 +2 r − 2 i + j 2 − 1 if i ∈ 2 Z + 1 and i − j 2 ∈ { 1 − r, − r, . . . , 2 − 2 r } , − x r r + i − 5 2 if i ∈ 2 Z and j = i + 2 r − 3 , 0 otherwise . (6.5) E ij =      ± 1 if i = j − 1 ± 1 and i ∈ 2 Z , x r i − 1 if i = j − 1 and i ∈ 2 Z + 1 , 0 otherwise . (6.6) 55 F or instance for B 3 , they read ( T ij ) i,j ≥ 1 =          x 1 0 0 x 2 1 0 − x 2 2 0 − x 1 3 0 − 1 0 0 0 0 − x 3 5 / 2 0 0 0 0 1 0 x 1 2 0 x 2 3 0 − x 2 4 0 − x 1 5 · · · 0 0 0 0 0 0 − x 3 9 / 2 0 0 0 0 1 0 x 1 4 0 x 2 5 0 − x 2 6 . . . . . .          , (6.7) ( E ij ) i,j ≥ 1 =            0 x 3 0 0 0 0 0 0 0 1 0 − 1 0 0 0 0 0 0 x 3 2 0 0 0 0 0 0 1 0 − 1 0 · · · 0 0 0 0 0 x 3 4 0 0 0 0 0 0 1 0 . . . . . .            . (6.8) Let T | u → u + s be the overall s hift of the lower index x a k → x a k + s in T in accorda nce with (6.4). As is e v ident fr om this example, the quantit y x a k is contained in T | u → u + s at most once as its matrix elemen t for any 1 ≤ a ≤ r and k . F or e x ample, the shift s = 1 is needed to a ccommo date x 1 1 as the (1,1) element of T | u → u + s . In view of this, we employ the nota tion T m ( i, j, ± x a k ) to mea n the m by m sub-matrix of T | u → u + s , where s is chosen so that its ( i, j ) element b ecomes exactly ± x a k . F or example in (6.7), T 3 (1 , 1 , x 1 0 ) =   x 1 0 0 x 2 1 0 0 0 1 0 x 1 2   , T 3 (1 , 1 , x 1 1 ) =   x 1 1 0 x 2 2 0 0 0 1 0 x 1 3   , T 2 (1 , 2 , − x 3 5 / 2 ) =  0 − x 3 5 / 2 0 x 2 3  , T 2 (1 , 2 , − x 3 2 ) =  0 − x 3 2 0 x 2 5 / 2  . W e also use the similar notation E m ( i, j, ± x r k ). Now the result for B r is stated as Theorem 6.3 ([136]) . F or u nr estricte d T-system for B r , the fol lowing formula is valid: T ( a ) m ( u ) = de t  T 2 m − 1 (1 , 1 , x a − m +1 ) + E 2 m − 1 (1 , 2 , x r − m + r − a + 1 2 )  (1 ≤ a < r ) , T ( r ) m ( u ) = ( − 1) m ( m − 1) / 2 det  T m (1 , 2 , − x r − 1 − m 2 +1 ) + E m (1 , 1 , x r − m 2 + 1 2 )  . 6.3. T yp e C r . Here we introduce the inﬁnite dimensional matrix T by T ij =        x j − i +1 i + j 2 − 1 if i − j ∈ { 1 , 0 , . . . , 1 − r } , − x i − j +2 r +1 i + j 2 − 1 if i − j ∈ { − 1 − r , − 2 − r, . . . , − 1 − 2 r } , 0 otherwise . (6.9) 56 F or instance, for C 2 , it reads ( T ij ) i,j ≥ 1 =         x 1 0 x 2 1 / 2 0 − x 2 3 / 2 − x 1 2 − 1 0 0 1 x 1 1 x 2 3 / 2 0 − x 2 5 / 2 − x 1 3 − 1 0 · · · 0 1 x 1 2 x 2 5 / 2 0 − x 2 7 / 2 − x 1 4 − 1 0 0 1 x 1 3 x 2 7 / 2 0 − x 2 9 / 2 − x 1 5 . . . . . .         . W e keep the notatio n (6.4) a nd T m ( i, j, ± x a k ) (1 ≤ a ≤ r ) as in Section 6 .2. Note that T m (1 , 2 , − x r k ) is an an ti-symmetric matrix for any m . Theorem 6. 4 ([136]) . F or unre stricte d T-system for C r , the fol lowing formula is valid: T ( a ) m ( u ) = de t T m (1 , 1 , x a − m 2 + 1 2 ) (1 ≤ a < r ) , (6.10) T ( r ) m ( u ) = ( − 1) m pf T 2 m (1 , 2 , − x r − m +1 ) . (6.11) As an additional result, we ha ve the following relations. T ( r ) m ( u − 1 2 ) T ( r ) m ( u + 1 2 ) = det T 2 m (1 , 1 , x r − m + 1 2 ) , (6.12) T ( r ) m ( u ) T ( r ) m +1 ( u ) = det T 2 m +1 (1 , 1 , x r − m ) . (6.13) If one extends the deﬁnition of x a k (6.4) by x a k + x 2 r + 2 − a k = 0 in accor dance with (9.31), then (6.10) is identical with the r e sult (6.1) for A 2 r + 1 . As remarked in the end of Section 2.1, the T-systems for B 2 and C 2 are equiv alent by the in ter change T (1) m ( u ) ↔ T (2) m ( u ). Therefore Theorems 6.3 and 6 .4 supply these T-systems with t wo kinds of Jacobi-T rudi type for m ulas. 6.4. T yp e D r . Here we deﬁne the inﬁnite dimensional matrices T and E b y T ij =                    x j − i 2 +1 i + j 2 − 1 if i ∈ 2 Z + 1 and i − j 2 ∈ { 1 , 0 , . . . , 3 − r } , − x r − 1 i + j − 1 2 if i ∈ 2 Z + 1 and i − j 2 = 5 2 − r , − x r i + j − 3 2 if i ∈ 2 Z + 1 and i − j 2 = 3 2 − r , − x i − j 2 +2 r − 3 i + j 2 − 1 if i ∈ 2 Z + 1 and i − j 2 ∈ { 1 − r , − r, . . . , 3 − 2 r } , 0 otherwise . (6.14) E ij =          ± 1 if i = j − 2 ± 2 and i ∈ 2 Z , x r − 1 i if i = j − 3 and i ∈ 2 Z , x r i − 2 if i = j − 1 and i ∈ 2 Z , 0 otherwise . (6.15) 57 F or instance for D 4 , they read ( T ij ) i,j ≥ 1 =        x 1 0 0 x 2 1 − x 3 2 0 − x 4 2 − x 2 3 0 − x 1 4 0 − 1 0 0 0 0 0 0 0 0 0 0 0 · · · 1 0 x 1 2 0 x 2 3 − x 3 4 0 − x 4 4 − x 2 5 0 − x 1 6 0 0 0 0 0 0 0 0 0 0 0 . . . . . .        , ( E ij ) i,j ≥ 1 =        0 0 0 0 0 0 0 0 0 0 1 x 4 0 0 x 3 2 − 1 0 0 0 . . . 0 0 0 0 0 0 0 0 0 0 0 0 1 x 4 2 0 x 3 4 − 1 0 . . .        . W e keep the notations (6.4), T m ( i, j, ± x a k ) (1 ≤ a ≤ r − 2) a nd T m ( i, j, − x a k ) , E m ( i, j, x a k ) ( a = r − 1 , r ) as in Section 6.2. Theorem 6.5 ([136]) . F or unr estricte d T-system for D r , the fol lowing formula is valid: T ( a ) m ( u ) = de t  T 2 m − 1 (1 , 1 , x a − m +1 ) + E 2 m − 1 (2 , 3 , x r − m − r + a +4 )  (1 ≤ a ≤ r − 2) , (6.16) T ( r − 1) m ( u ) = pf  T 2 m (2 , 1 , − x r − 1 − m +1 ) + E 2 m (1 , 2 , x r − 1 − m +1 )  , (6.1 7) T ( r ) m ( u ) = ( − 1) m pf  T 2 m (1 , 2 , − x r − m +1 ) + E 2 m (2 , 1 , x r − m +1 )  . (6.18) The matric e s in (6.1 7) a nd (6.18) are indeed anti-symmetric. T he following relations also hold. T ( r − 1) m ( u ) T ( r ) m ( u ) = ( − 1) m det  T 2 m (1 , 1 , − x r − 1 − m +1 ) + E 2 m (2 , 2 , x r − m +1 )  , T ( r − 1) m ( u + 1) T ( r ) m ( u − 1) = ( − 1) m det  T 2 m (1 , 1 , − x r − m ) + E 2 m (2 , 2 , x r − 1 − m +2 )  , T ( r − 1) m +1 ( u ) T ( r ) m ( u − 1) = ( − 1) m +1 det  T 2 m +1 (1 , 1 , − x r − 1 − m ) + E 2 m +1 (2 , 2 , x r − m )  , T ( r − 1) m ( u + 1) T ( r ) m +1 ( u ) = ( − 1) m det  T 2 m +1 (2 , 1 , x r − 2 − m +1 ) + E 2 m +1 (1 , 1 , x r − m )  . Theorems 6.3 – 6.5 can only b e prov ed b y using (6.2) and the fac t (pf ) 2 = det. 6.5. Another Jacobi-T rudi ty p e formula for B r . F or B r and D r , a v aria nt of the Jacobi-T rudi t yp e formula is known whic h has a quite similar str ucture to the A r case. Compared with the rather sparse matrices T and E , the r elev ant matrices are dense and involv e so me auxiliary v ariables. Here we pre s ent the result for B r . The D r case is similar although slightly mo re inv olved. Given T (1) 1 ( u ) , . . . , T ( r ) 1 ( u ), we introduce the auxiliary v ariable T a ( u ) fo r all a ∈ Z by T a ( u ) =      0 a < 0 , 1 a = 0 , T ( a ) 1 ( u ) 1 ≤ a ≤ r − 1 , (6.19) T a ( u ) + T 2 r − 1 − a ( u ) = T ( r ) 1 ( u − r + a + 1 2 ) T ( r ) 1 ( u + r − a − 1 2 ) for all a ∈ Z . (6.20) Recall that t a = 1 for a 6 = r a nd t r = 2 for B r according to (2.1). 58 Theorem 6.6 ([137]) . F or u nr estricte d T-system for B r , the fol lowing formula is valid: T ( a ) t a m ( u ) = de t ( T a + i − j ( u + i + j − m − 1)) 1 ≤ i,j ≤ m (1 ≤ a ≤ r ) , (6.21) T ( r ) 2 m +1 ( u ) =          T ( r ) 1 ( u − m ) T r − 1 ( u − m + 1 2 ) T r − 2 ( u − m + 3 2 ) · · · T r − m ( u − 1 2 ) T ( r ) 1 ( u − m + 2) T r ( u − m + 3 2 ) T r − 1 ( u − m + 5 2 ) · · · T r − m +1 ( u + 1 2 ) . . . . . . . . . . . . . . . T ( r ) 1 ( u + m ) T r + m − 1 ( u + 1 2 ) T r + m − 2 ( u + 3 2 ) · · · T r ( u + m − 1 2 )          , (6.22) wher e t he matrix (6.22) is of size m + 1 , its ( i + 1 , 1) element is T ( r ) 1 ( u − m + 2 i ) and t he r est has the same p att ern as (6.21) for T ( r ) 2 m +2 ( u − 1 2 ) . 6.6. Bibliog raphical n o tes. The formulas (6.1)–(6.3) for A r in Theo rem 6.1 ﬁrst app eared in [59] befo r e the T-system was formulated. T he r e, transfer matrices more g eneral tha n T ( a ) m ( u ) were considered. Theorems 6 .3 – 6.5 supplemented the determinant co njectur e s in [1] with Pfaﬃans. A result for D r analogo us to Theorem 6.6 is av aila ble in [1 38]. 7. T ableau sum formula 7.1. T yp e A r . Let 1 u , . . . , r + 1 u be v aria bles dep ending on u . If we s e t T (1) 1 ( u ) = P r +1 a =1 a u , then T (1) 1 ( u − 1) T (1) 1 ( u + 1) = X a ≤ b a u − 1 b u +1 + X a>b b u +1 a u − 1 , (7.1) where the b oth arrays of the b oxes stand for the pr o duct. Comparing this with the T-sy stem re la tion T (1) 1 ( u − 1) T (1) 1 ( u + 1) = T (1) 2 ( u ) + T (2) 1 ( u ), one may identif y T (1) 2 ( u ) and T (2) 1 ( u ) individually with the tw o terms in (7.1), and try to further establish similar for mu las for higher T ( a ) m ( u ). Such a pro cedure leads to a so lution of the T-system expr essed as a sum of ta blea ux. In fac t, if one for gets the sp ec- tral para meter u in (7.1), it can b e viewed as the identit y among Sch ur functions corres p o nding to the irr educible decomp osition of the A r -mo dules: ⊗ = ⊕ (7.2) In this sense the result pr e sented in what follows for A r is a defor mation of the classical tableau sum formula for the Sch ur functions [1 35]. Consider the Y oung diagram ( m a ) of a × m rectangular shape . Let T ab( m a ) be the set o f semistandar d tablea ux o n ( m a ) with n umbers { 1 , 2 , . . . , r + 1 } . The inscrib ed num b ers a re strictly increasing to the b ottom and non-decr easing to the right. F or example when r = 2, T ab(2) = { } , , , , , , T ab(2 2 ) = n 1 1 1 2 1 3 2 2 2 3 3 3 o . , , , , , 1 1 1 1 1 1 1 2 1 2 2 2 2 2 2 3 3 3 2 3 3 3 3 3 59 Note that T ab( m a ) is empty for a > r + 1. W e deﬁne T u = a Y i =1 m Y j =1 t ij u + a − m − 2 i +2 j for T = ( t ij ) ∈ T ab( m a ) , (7.3) where t ij denotes the en try of the b ox in the i th r ow and the j th column from the top left. Theorem 7. 1. T ( a ) m ( u ) = X T ∈ T ab( m a ) T u (1 ≤ a ≤ r + 1) (7.4) is a solut ion of the T-system for A r (2.5). W e note that T ( r +1) m ( u ) her e is not just 1 but non tr ivially c hos en as (7.4) as opp osed to the or iginal deﬁnition of the T-sy stem. How ever, T ab( m r +1 ) consists of a unique tableau; therefore, (7.4) states that T ( r +1) m ( u ) is a monomial: T ( r +1) m ( u ) = m Y j =1 T ( r +1) 1 ( u − m − 1 + 2 j ) , T ( r +1) 1 ( u ) = r +1 Y i =1 i u + r +2 − 2 i . (7.5) Thu s the s ituation T ( r +1) m ( u ) = 1 can be res tored if the v aria bles 1 u , . . . , r + 1 u are chosen so as to satisfy the simple relation T ( r +1) 1 ( u ) = 1. Theorem 7.1 yields the q -characters b y the sp ecial choice a u = z a ( u ) := Y − 1 a − 1 ,q u + a Y a,q u + a − 1 ( Y 0 ,q u = Y r +1 ,q u = 1) , (7.6) which indeed s a tisﬁes the condition T ( r +1) m ( u ) = 1. The restriction (4.22)-(4.23) o f the resulting q -character T ( a ) m ( u ) = χ q ( W ( a ) m ( u )) is given by res T ( a ) m ( u ) = χ ( V mω a ) (7.7) in the notatio n of (4.24) since the a × m r ectangle Y oung diagr am corr esp onds to the highest weigh t mω a . In the rest of this sectio n we s ha ll present the tablea u sum formulas fo r g = B r , C r , D r along the context of the q - characters T ( a ) m ( u ) = χ q ( W ( a ) m ( u )). The con- ten ts cov er a ll the fundamental ones T (1) 1 ( u ) , . . . , T ( r ) 1 ( u ), which is enoug h in prin- ciple to determine all the higher ones T ( a ) m ( u ) due to Theorem 4.3. Some T ( a ) m ( u ) allowing a relatively simple description will also b e included. 7.2. T yp e B r . Let us introduce the index set a nd a total order on it as J = { 1 , 2 . . . , r, 0 , r, . . . , 2 , 1 } , 1 ≺ · · · ≺ r ≺ 0 ≺ r ≺ · · · ≺ 1 . (7.8) W e introduce the v aria bles co rresp onding to single b ox tableaux . z a ( u ) = Y a,q 2 u +2 a − 2 Y − 1 a − 1 ,q 2 u +2 a (1 ≤ a ≤ r − 1) , z r ( u ) = Y r,q 2 u +2 r − 3 Y r,q 2 u +2 r − 1 Y − 1 r − 1 ,q 2 u +2 r , z 0 ( u ) = Y r,q 2 u +2 r − 1 Y r,q 2 u +2 r − 3 Y − 1 r,q 2 u +2 r +1 Y − 1 r,q 2 u +2 r − 1 , z r ( u ) = Y r − 1 ,q 2 u +2 r − 2 Y − 1 r,q 2 u +2 r − 1 Y − 1 r,q 2 u +2 r +1 , z a ( u ) = Y a − 1 ,q 2 u +4 r − 2 a − 2 Y − 1 a,q 2 u +4 r − 2 a (1 ≤ a ≤ r − 1) , (7.9) 60 where Y 0 ,q k = 1. ( z 0 ( u ) in p1 427 of [139] co ntains a mispr int .) Consider the Y oung diagram ( m a ) of a × m rectangular shap e. Let T ab( B r , m a ) be the se t of tableaux on ( m a ) with e n tries from J . The letter t i,j ∈ J inscrib ed on the i th row and the j th column from the top left corner should satisfy the following conditions for any adjacent pair: t i,j  t i,j +1 and ( t i,j , t i,j +1 ) 6 = (0 , 0) , t i,j ≺ t i +1 ,j or ( t i,j , t i +1 ,j ) = (0 , 0) . (7.10) Given a tableau T = ( t i,j ) ∈ T a b( B r , m a ) we set T u = a Y i =1 m Y j =1 z t i,j ( u + a − m + 2 i + 2 j ) . (7.11) This is an analog of the A r case (7.3). Theorem 7. 2 ([13 7, 68]) . The q -char acter T ( a ) t a m ( u ) = χ q ( W ( a ) t a m ( u )) is given by T ( a ) t a m ( u ) = X T ∈ T ab ( B r ,m a ) T u (1 ≤ a ≤ r ) . (7.12) Recall that t a (2.1) is 1 exce pt t r = 2 for B r . The formula (7.12) is rela ted to (6.21) in a pa rallel wa y with the A r case explaine d in the pr evious subsection. A similar r esult is av a ilable for the r emaining ca s e T ( r ) 2 m +1 ( u ) ba sed on (6 .22) [137]. Theorem 7 .2 follows by combining the fa cts that the RHS and the T ( r ) 2 m +1 ( u ) s a tisfy the T-system [13 7], q -characters also satisfy the T-system [68], and the T ( a ) m ( u ) is uniquely determined by the T-sy stem and T ( a ) 1 ( u ) ( a ∈ I ). See also [140]. Here we only give the formula for T ( r ) 1 ( u ). It is known that the U q ( B (1) r )-mo dule W ( r ) 1 ( u ) is isomorphic as a U q ( B r )-mo dule to the spin representation of the latter. Its w eights are m ultiplicity-free and natur ally la be le d with the arrays ( σ 1 , . . . , σ r ) ∈ {± 1 } r . According ly w e intro duce ( σ 1 , . . . , σ r ) u = r Y a =1  Y a,q 2 u +2 r − 1 − ρ a  1 2 ( σ a − σ a +1 ) , (7.13) ρ a = 2( σ 1 + · · · + σ a − 1 ) + σ a − σ a +1 t a , σ r +1 = − σ r . (7.14) Then w e hav e T ( r ) 1 ( u ) = X σ 1 ,...,σ r = ± 1 ( σ 1 , . . . , σ r ) u . (7 .15) F or r = 2, T (2) 1 ( u ) = χ q ( W (2) 1 ( u )) has b een written down in Example 4.5. 7.3. T yp e C r . Let us introduce the index set a nd a total order on it as J = { 1 , 2 . . . , r, r, . . . , 2 , 1 } , 1 ≺ · · · ≺ r ≺ r ≺ · · · ≺ 1 . (7 .16) F or 1 ≤ a ≤ r we set z a ( u ) = Y a,q 2 u + a − 1 Y − 1 a − 1 ,q 2 u + a , z a ( u ) = Y a − 1 ,q 2 u +2 r − a +2 Y − 1 a,q 2 u +2 r − a +3 , (7.17) 61 where Y 0 ,q k = 1. Here we present the tableau s um formulas for T (1) m ( u ) and T ( a ) 1 ( u ). Consider the Y oung diagra m ( m ) with length m one row sha pe. Let T ab( C r , ( m )) be the set of tableaux on it with entries fr om J having the following for m: 2 n z }| { i 1 · · · i k r r · · · r r j l · · · j 1 1  i 1  · · ·  i k  r , r  j l  · · ·  j 1  1. (7.18) Here k , l a nd n a re any nonnega tive integers sa tis fying k + 2 n + l = m . Let thos e tableaux b e denoted simply by the ar ray o f ent ries as ( i 1 , . . . , j 1 ) ∈ J m . Then we hav e T (1) m ( u ) = X ( i 1 ,...,i m ) ∈ T ab( C r , ( m )) m Y k =1 z i k  u + 2 k − m − 1 2  . (7.19) Consider the Y oung diagram (1 a ) with length a one column sha pe. Let T a b( C r , (1 a )) be the set of tableaux on it with entries from J . The letter i k ∈ J inscribed on the k th row from the top sho uld satisfy the conditions: i 1 ≺ · · · ≺ i a , r + k − l ≥ c for any k, l , c such that i k = c, i l = c. (7.20) Denote suc h a tableau by the arr ay ( i 1 , . . . , i a ) ∈ J a . Then we have T ( a ) 1 ( u ) = X ( i 1 ,...,i a ) ∈ T ab( C r , (1 a )) a Y k =1 z i k  u + a + 1 − 2 k 2  (1 ≤ a ≤ r ) . (7.21) W e note that T (1) m ( u ) and T ( a ) 1 ( u ) a re the s implest cases in that the tablea u rules can actually b e describ ed just by arrays without introducing a tableau. 7.4. T yp e D r . Here we treat T (1) m ( u ) and the fundamental q -character s T ( a ) 1 ( u ). Let us in tro duce the index set and a partial or der on it a s J = { 1 , 2 . . . , r, r , . . . , 2 , 1 } , 1 ≺ · · · ≺ r − 1 ≺ r r ≺ r − 1 ≺ · · · ≺ 1 , (7.22) where no order is assumed b etw een r and r . F o r i ∈ J , deﬁne z i ( u ) by z a ( u ) = Y a,q u + a − 1 Y − 1 a − 1 ,q u + a (1 ≤ a ≤ r − 2) , z r − 1 ( u ) = Y r − 1 ,q u + r − 2 Y r,q u + r − 2 Y − 1 r − 2 ,q u + r − 1 , z r ( u ) = Y r,q u + r − 2 Y − 1 r − 1 ,q u + r , z r ( u ) = Y r − 1 ,q u + r − 2 Y − 1 r,q u + r , z r − 1 ( u ) = Y r − 2 ,q u + r − 1 Y − 1 r − 1 ,q u + r Y − 1 r,q u + r , z a ( u ) = Y a − 1 ,q u +2 r − a − 2 Y − 1 a,q u +2 r − a − 1 (1 ≤ a ≤ r − 2) , (7.23) where Y 0 ,q k = 1. Let T a b( D r , ( m )) b e the set of one r ow tableaux ( i 1 , . . . , i m ) ∈ J m ob eying the condition: i 1 ≺ · · · ≺ i m , r and r do not a ppe a r simultaneously . (7.24) 62 Then w e hav e T (1) m ( u ) = X ( i 1 ,...,i m ) ∈ T ab( D r , ( m )) m Y k =1 z i k ( u + 2 k − m − 1) . (7.2 5 ) F or 1 ≤ a ≤ r − 2, let T ab( D r , (1 a )) b e the set of o ne column tableaux ( i 1 , . . . , i a ) ∈ J a ob eying the condition: i k ≺ i k +1 or ( i k , i k +1 ) = ( r, r ) or ( i k , i k +1 ) = ( r , r ) for 1 ≤ k ≤ a − 1 . (7.26) Then w e hav e T ( a ) 1 ( u ) = X ( i 1 ,...,i a ) ∈ T ab( D r , (1 a )) a Y k =1 z i k ( u + a + 1 − 2 k ) (1 ≤ a ≤ r − 2) . (7.27) It is known that the U q ( D (1) r )-mo dules W ( r − 1) 1 ( u ) and W ( r ) 1 ( u ) are isomor phic as U q ( D r )-mo dules to the spin repres ent ations o f the latter. T heir weigh ts are m ultiplicit y -free and na tur ally la b e le d with the a rrays ( σ 1 , . . . , σ r ) ∈ {± 1 } r . Ac- cordingly we introduce ( σ 1 , . . . , σ r ) u =  Y r,q u + r − 1 − ρ r  1 2 ( σ r + σ r − 1 ) r − 1 Y a =1  Y a,q u + r − 1 − ρ a  1 2 ( σ a − σ a +1 ) , (7.28) ρ a = ( σ 1 + · · · + σ a − 1 + σ a − σ a +1 2 1 ≤ a ≤ r − 1 , σ 1 + · · · + σ r − 2 + σ r + σ r − 1 2 a = r . (7.29) It follows that ( σ 1 , . . . , σ r − 1 , − σ r ) u = ( σ 1 , . . . , σ r ) u | Y r,q k ↔ Y r − 1 ,q k . (7.30 ) W e hav e T ( r − 1) 1 ( u ) = X σ 1 ,...,σ r = ± 1 σ 1 ··· σ r = − 1 ( σ 1 , . . . , σ r ) u , T ( r ) 1 ( u ) = X σ 1 ,...,σ r = ± 1 σ 1 ··· σ r =1 ( σ 1 , . . . , σ r ) u . (7.31) 7.5. Bibliog raphical no tes. T ableau sums in Theorems 7.1 and 7.2 w er e resp ec- tively given in [59] a nd [137] in the context of analytic B e the ansatz for more gener al skew shap e Y oung diagra ms . A uniform pr o of of the equa lit y b etw een the Jac o bi- T rudi t yp e determinant and the ta ble a u sum is av ailable in [1 41]. F or t y pe A r , see also [142] for an account fro m the viewp o int o f Macdonald’s ninth v ar iation of Sch ur functions [1 43]. The tableau sums in Sections 7 .3 a nd 7.4 ﬁrst app eared in the a na- lytic Bethe ansatz [1 4 4]. The sums o f the sa me structur e are used in the deformed W -a lgebras [1 45]. T ableau c o nstructions of higher T ( a ) m ( u ) for C r and D r , whic h are signiﬁcantly more involv ed than A r and B r , hav e b een achiev ed in [1 46, 140]. In this section we ha ve only treated the unt wisted case U q ( ˆ g ). F or tableau sum formulas for T-systems in twisted c ase, see [12, 13] and reference therein. 8. Anal ytic Bethe ansa tz Let T ( a ) m ( u ) be the commuting tr ansfer matrix of a s olv able lattice mode l in the sense of Section 3 . Ther e is an empirica l method called analytic Bethe ansatz to pro duce eigen v alues of T ( a ) m ( u ) in many cases. Those eigenv a lue formulas p osse s s a sp eciﬁc “dress e d v acuum form” which necessa rily satisfy the T-sys tem in Remark 2.7 with a nontrivial g ( a ) m ( u ). Here we consider the Bethe equation and dresse d 63 v acuum forms for general g and T ( a ) m ( u ), and reformulate the co nv entional ana ly tic Bethe ansatz via its connection with q -character s . 8.1. A 1 case. Cons ider the 6 vertex mo del (3.1). Here we employ the normaliz a - tion 1 1 1 1 2 2 2 2 1 2 2 1 2 1 1 2 2 1 2 1 1 2 1 2 [2 + u ] q 1 / 2 [2 + u ] q 1 / 2 [ u ] q 1 / 2 [ u ] q 1 / 2 z 1 / 2 [2] q 1 / 2 z − 1 / 2 [2] q 1 / 2 , (8.1) which is o btained by dividing (3.1) by ( z q ) 1 / 2 (1 − q ) and setting z = q u . F or the deﬁnition o f the sym b ol [ u ] q , see (3.18). Let T 1 ( u ) b e the tra ns fer matrix (3 .1 1) with m = 1 a nd w j = q v j . Its eig e n v alue (deno ted by the same symbol) is given by [2] T 1 ( u ) = 1 u + 2 u , (8.2) 1 u = φ ( u + 2) Q ( u − 1) Q ( u + 1) , 2 u = φ ( u ) Q ( u + 3) Q ( u + 1) . (8.3) Here φ ( u ) = Q N j =1 [ u − v j ] q 1 / 2 and Q ( u ) = Q 1 ( u ) is called Baxter’s Q - function Q ( u ) = Q n j =1 [ u − u j ] q 1 / 2 with u 1 , . . . , u n determined from the Bethe equa tio n − φ ( u j + 1) φ ( u j − 1) = Q ( u j + 2) Q ( u j − 2) (1 ≤ j ≤ n ) . (8.4) Here, n is the num b er of down spins preser ved under T 1 ( u ). The factors φ ( u + 2) and φ ( u ) in (8 .3) are ca lled v acuum parts in the s ense that they ar e already pr e sent in the v ac uum secto r n = 0 where Q ( u ) = 1. In fact, the vector 11 . . . 1 is obviously the unique eigenv ector with the v acuum eigenv alue: N Y j =1 [ u − v j + 2] q 1 / 2 + N Y j =1 [ u − v j ] q 1 / 2 = φ ( u + 2) + φ ( u ) . (8.5) The factors inv olving Q -functions in (8 .3) are called dr ess par ts, and the eigenv a lue formula of the form (8.2)–(8.3) is called a dre s sed v acuum form. The v acuum part is non- universal in that it is directly aﬀected by the no rmalization of the Bo ltzmann weigh ts (relev a nt R matrix ) and also dep ends on the q uantum space data such as inhomogeneity { v j } entering φ ( u ). On the other hand, the dr ess par t enco des the structure of the auxiliary space essentially as w e will se e b elow. The dres sed v acuum form has an apparent p ole a t u = − 1 + u j bec ause of Q ( u j ) = 0. The Bethe equa tion (8.4) tells that it is actually spurious provided that u j is distinct from the o ther ro ots. This is compatible with the prop erty that eigenv alue s of the transfer matrix are reg ular functions of u if the loc a l Bo ltzmann weigh ts are so. The analytic Bethe ansatz is a hypothesis that one c an reverse these arguments to repro duce the eig env alue formula fro m its characteristic prop er ties bypassing the construction of eigenv ecto rs. O ne sta rts with the ansatz dress ed v acuum form with 64 the prescrib ed v acuum part T 1 ( u ) = φ ( u + 2) Q ( u + a ) Q ( u + b ) + φ ( u ) Q ( u + c ) Q ( u + d ) . (8.6) Then a, b, c , d a re determined by dema nding that the p ole- freeness is formally g uar- anteed by the Bethe equatio n (8.4) which one somehow admits from the ons et. In the present example, this certainly ﬁxes a, b, c, d uniquely as in (8.3). F urther sup- plement ary conditions may als o be tak en into account such as asy mptotic b ehavior as | u | → ∞ and the symmetry under complex conjuga tion, etc. It is not known whether such a pro cedure indeed lea ds to the unique and cor rect eig env alue for mula in genera l. Instead we sha ll prop ose in Section 8.2 a constructive wa y of pro ducing the dressed v acuum form for general U q ( ˆ g ) by utilizing q -characters. In the remainder of this subsectio n, we illustrate the simplest solution o f the T-system for A 1 in the dressed v acuum form. Although the result is obtainable by sp ecializing the tableau sum for mula (7.3), w e re-derive it here for later conv enience. F or simplicit y T (1) m ( u ) w ill b e denoted by T m ( u ). Then the pro duct o f (8.2) is written as T 1 ( u − 1) T 1 ( u + 1) = 1 u − 1 1 u +1 + 1 u − 1 2 u +1 + 2 u − 1 2 u +1 + 1 u +1 2 u − 1 . By (8.3 ), the last term b ecomes φ ( u − 1 ) φ ( u + 3), which is indep endent of Q ( u ). Ident ifying the other three terms with T 2 ( u ), one has T 1 ( u − 1) T 1 ( u + 1) = T 2 ( u ) + φ ( u − 1) φ ( u + 3) , which is an a ﬃnization o f the identit y (doublet ) ⊗ 2 = (triplet) ⊕ (sing le t) depicted as (7.2). It is eas y to systematize this calcula tio n to s how that T m ( u ) = X 1 ≤ i 1 ≤···≤ i m ≤ 2 i 1 u − m +1 i 2 u − m +3 · · · i m u + m − 1 (8.7) is a solution of the unrestricted T-system for A 1 on the eigenv alues: T m ( u − 1) T m ( u + 1) = T m − 1 ( u ) T m +1 ( u ) + g m ( u ) , (8.8) g m ( u ) = m − 1 Y k =0 φ ( u + 2 k − m ) φ ( u + 4 + 2 k − m ) . (8.9 ) Explicitly , (8.7) reads as T m ( u ) =  m − 1 Y k =1 φ ( u + m + 1 − 2 k )  m X j =0 Q ( u − m ) Q ( u + m + 2) φ ( u + m + 1 − 2 j ) Q ( u + m − 2 j ) Q ( u + m + 2 − 2 j ) . (8.10) The summands in (8.7) a re naturally labeled with the semistandard tableaux of length m row shap e ( m ) on num b ers { 1 , 2 } . Note that g m ( u − 1) g m ( u + 1) = g m − 1 ( u ) g m +1 ( u ) (8.11) is satisﬁed with g 0 ( u ) = 1. Although the explicit form (8.10) is not par ticularly more illuminating than (8.7), one can ea s ily chec k that it is forma lly p o le-free in the sa me ma nner as b efore thanks to the Bethe equation (8.4). Another wa y of 65 seeing this is of cours e by the Jacobi-T r udi type formula (6.1) with r = 1 mo diﬁed as T (2) 1 ( u ) = g 1 ( u ), e.g. T 3 ( u ) =       T 1 ( u − 2) g 1 ( u − 1) 0 1 T 1 ( u ) g 1 ( u + 1) 0 1 T 1 ( u + 2)       . Thu s the po le-freeness of T m ( u ) is an obvious co rollary of that for T 1 ( u ). 8.2. Dressed v acuum form and q -cha racters. The ana lytic Bethe ansa tz is ex- tended to the genera l U q ( ˆ g ) and further sharp ened by a connection with the theory of q -characters. Fir st we mak e a motive observ ation on the simplest exa mple. Re- call the q -character o f W (1) 1 ( u ), the “spin 1 2 representation” of U q ( A (1) 1 ) in Exa mple 4.4: χ q ( W (1) 1 ( u )) = Y z + Y − 1 z q 2 ( z = q u ) . (8.12) On the other hand, the dressed v acuum form (8.2)–(8.3) of the 6-vertex mo del transfer matrix reads T (1) 1 ( u ) = φ ( u + 2) Q ( u − 1) Q ( u + 1) + φ ( u ) Q ( u + 3) Q ( u + 1) . (8 .13) Upo n substitution Y q u → η ( u − 1) Q ( u − 1 ) η ( u + 1) Q ( u + 1 ) , the q -character (8 .1 2) bec o mes η ( u − 1) η ( u + 1) Q ( u − 1) Q ( u + 1) + η ( u + 3) η ( u + 1) Q ( u + 3) Q ( u + 1) . Thu s the ab ov e substitution with the following overall renorma lization φ ( u + 2) η ( u + 1) η ( u − 1) χ q ( W (1) 1 ( u )) = φ ( u + 2) Q ( u − 1) Q ( u + 1) + φ ( u + 2) η ( u + 3) η ( u − 1) Q ( u + 3) Q ( u + 1) repro duces the dressed v acuum form (8.13) if η ( u ) is a ssumed to ob ey the diﬀerence equation φ ( u + 1) φ ( u − 1) = η ( u − 2) η ( u + 2) . (8.14) Note that this equation has the form of the Bethe eq uation (8.4): − φ ( u j + 1) φ ( u j − 1) = Q ( u j + 2) Q ( u j − 2) without the sign facto r, a nd Q and u j being repla ced by η − 1 and u , resp ectively . The same feature will be adopted in (8.19). The connection o f (8.12) a nd (8.13) originates in the fact that the former is the q -character of W (1) 1 ( u ) which is the auxiliary space of the transfer matrix relev a nt to the la tter. Now we g eneralize these observ ations to U q ( ˆ g ). Consider the trigonometric vertex mo del asso ciated with U q ( ˆ g ) under the perio dic bo undary condition. Let T ( a ) m ( u ) be the transfer ma trix (3.44) with the auxiliary space W ( a ) m ( u ) and the quantum space W ( r 1 ) s 1 ( v 1 ) ⊗ · · · ⊗ W ( r N ) s N ( v N ): T ( a ) m ( u ) = T r W ( a ) m ( u )  R ( a,m ; r N ,s N ) 0 ,N ( z / w N ) · · · R ( a,m ; r 1 ,s 1 ) 0 , 1 ( z / w 1 )  ∈ End( W ( r 1 ) s 1 ( v 1 ) ⊗ · · · ⊗ W ( r N ) s N ( v N )) , (8.15) 66 where z = q tu , w i = q tv i . Due to the Y ang-Ba xter equa tion, they are c ommut ative, i.e. [ T ( a ) m ( u ) , T ( b ) n ( v )] = 0. The pr o blem is to ﬁnd their join t sp ectrum. Let us construct a r elev ant dressed v acuum form Λ ( a ) m ( u ) for T ( a ) m ( u ). In the following, a simple identit y A a,z | Y c,z → f c ( u − 1 /t c ) f c ( u +1 /t c ) = r Y b =1 f b ( u − ( α a | α b )) f b ( u + ( α a | α b )) ( z = q tu ) (8.16) for any functions f 1 , . . . , f r will b e useful. See (2.1) and (4.25) for the deﬁnitions of t a , t a nd A a,z . First we intro duce an “unnormalized” dressed v acuum form: ˜ Λ ( a ) m ( u ) = χ q ( W ( a ) m ( u )) with subs titution Y c,q tv → η c ( v − 1 t c ) η c ( v + 1 t c ) Q c ( v − 1 t c ) Q c ( v + 1 t c ) . (8.17) Let ˜ A c,q tv be the result of the same s ubs titution into A c,q tv . By the deﬁnition we hav e ˜ Λ ( a ) m ( u ) = η a ( u − m t a ) η a ( u + m t a ) Q a ( u − m t a ) Q a ( u + m t a ) 1 + X c,v monomial in ˜ A − 1 c,q tv ! . (8.18 ) Here the factor ( η a Q a ) / ( η a Q a ) is the top term sp eciﬁed by (4.21) and (8.17). The app earance of ˜ A − 1 c,q tv is due to Theo rem 4.6 (1). As for the functions η 1 , . . . , η r , we po stulate, as the generaliza tion o f (8.14), the following diﬀerence equatio n N Y k =1 r k = a  u − v k + s k t a  q t/ 2  u − v k − s k t a  q t/ 2 = r Y b =1 η b ( u − ( α a | α b )) η b ( u + ( α a | α b )) (1 ≤ a ≤ r ) , (8.19) where [ u ] p is deﬁned in (3 . 18). Then using (8.16) and (8.19) we ﬁnd ˜ A a,q tu = r Y b =1 η b ( u − ( α a | α b )) Q b ( u − ( α a | α b )) η b ( u + ( α a | α b )) Q b ( u + ( α a | α b )) = N Y k =1 r k = a  u − v k + s k t a  q t/ 2  u − v k − s k t a  q t/ 2 · r Y b =1 Q b ( u − ( α a | α b )) Q b ( u + ( α a | α b )) . (8.20) Next we a djust the ov era ll normaliz a tion. Consider the R matrix on W ( a ) m ( u ) ⊗ W ( b ) s ( v ) and write its unique diagona l matrix element b etw een the tensor pro duct of the highest weigh t vectors a s φ ( a,b ) m,s ( u − v ). Namely , φ ( a,b ) m,s ( u − v ) = B o ltzmann w eight of the vertex mω a mω a . sω b sω b (8.21) Now we deﬁne the no rmalized dressed v acuum form by Λ ( a ) m ( u ) = N Y k =1 φ ( a,r k ) m,s k ( u − v k ) ! η a ( u + m t a ) η a ( u − m t a ) ˜ Λ ( a ) m ( u ) = N Y k =1 φ ( a,r k ) m,s k ( u − v k ) ! Q a ( u − m t a ) Q a ( u + m t a ) 1 + X c,v monomial in ˜ A − 1 c,q tv ! . (8.22) 67 Besides the (in pr inciple) k nown Boltzmann weigh ts φ ( a,b ) m,k , this only contains the Q -functions Q 1 , . . . , Q r and the LHS of (8.19). Recall tha t the trans fer matrices pre serve the subspa ces (sector s ) of the q uantum space sp eciﬁed by the weigh t. Let us par ameterize the weigh t by the nonnega tive int egers n 1 , . . . , n r as N X k =1 s k ω r k − r X a =1 n a α a , (8.2 3 ) where ω 1 , . . . , ω r denote the fundamental weight s of g (2.2). Given n a , we set Q a ( u ) = n a Y j =1 [ u − u ( a ) j ] q t/ 2 (8.24) by introducing the unknowns { u ( a ) j | 1 ≤ a ≤ r, 1 ≤ j ≤ n a } . Conjecture 8.1. L et T ( a ) m ( u ) (8.15) b e the t ra n sfer matrix normalize d as ( 8.21 ). Then its eigenvalues in the se ctor (8.23) ar e given by t he dr esse d vacuum form Λ ( a ) m ( u ) (8.22), (8.24) with the numb ers { u ( a ) j | 1 ≤ a ≤ r, 1 ≤ j ≤ n a } satisfying t he Bethe e qu ation: N Y k =1 r k = a  u ( a ) j − v k + s k t a  q t/ 2  u ( a ) j − v k − s k t a  q t/ 2 = − r Y b =1 Q b ( u ( a ) j + ( α a | α b )) Q b ( u ( a ) j − ( α a | α b )) . ( 8.25) Practica lly the res ults in Sectio n 7 serve as a large input to the pr escription (8.17) to produce Λ ( a ) m ( u ). The functions Q a ( u ) ar e called the (generalized) Baxter Q -functions. In view o f Theor em 4.6 (2), w e exp ect that their zero s, if in a generic po sition, do not cause a p ole in Λ ( a ) m ( u ) due to the Bethe equatio n. Let P a ( ζ ) b e the pro duct of the a th Drinfeld p olynomial (4.8) for each comp onent in the quant um space W ( r 1 ) s 1 ( v 1 ) ⊗ · · · ⊗ W ( r N ) s N ( v N ): P a ( ζ ) = N Y k =1 r k = a s k Y i =1 (1 − ζ q t ( v k +( s k +1 − 2 i ) /t a ) ) , deg P a = N X k =1 r k = a s k . (8.26) W e remar k that the LHS of (8.19) is expressed as N Y k =1 r k = a  u − v k + s k t a  q t/ 2  u − v k − s k t a  q t/ 2 = q deg P a a P a ( ζ q − 1 a ) P a ( ζ q a ) ( ζ = q − tu ) , (8.27) which further b e comes the LHS of the Bethe equation (8.2 5) b y the sp ecia lization u = u ( a ) j . This has forma lly the sa me for m a s (4.7). Note how ever that the quantum space W ( r 1 ) s 1 ( v 1 ) ⊗ · · · ⊗ W ( r N ) s N ( v N ) under consider a tion is not neces sarily ir reducible in g eneral, and the ab ove P a ( ζ ) is the a th Drinfeld p o lynomial of its irreducible quotient containing the tensor pr o duct of the highest weigh t v ecto rs. By the constr uc tio n (8.1 7) and Theorem 4.8, the unnorma lized dressed v a cuum form ˜ Λ ( a ) m ( u ) sa tis ﬁes the unrestricted T-s ystem for g . It follows that the nor malized one T ( a ) m ( u ) = Λ ( a ) m ( u ) (8 .22) sa tisﬁes the modiﬁed T-system containing an extra factor g ( a ) m ( u ) as (2.22): T ( a ) m ( u − 1 t a ) T ( a ) m ( u + 1 t a ) = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) + g ( a ) m ( u ) M ( a ) m ( u ) , 68 where the origina l T-sys tem corresp onds to g ( a ) m ( u ) = 1 a s in (2.18). The sca lar factor g ( a ) m ( u ) has the prop erties : (i)Apart from ( a, m, u ), it o nly depends o n the quantum spac e data W ( r 1 ) s 1 ( v 1 ) ⊗ · · · ⊗ W ( r N ) s N ( v N ). (ii) It satisﬁes relation (2.23): g ( a ) m ( u − 1 t a ) g ( a ) m ( u + 1 t a ) = g ( a ) m − 1 ( u ) g ( a ) m +1 ( u ) . In fact this ha s been encountered for g = A 1 in (8.11). T o de r ive these pr op erties, note that the fusion construction implies that the dia g onal element o f the R -matrix (8.21) is factorize d as φ ( a,b ) m,s ( u ) = Q m i =1 φ ( a,b ) 1 ,s ( u + ( m + 1 − 2 i ) /t a ). Thus the ﬁrst relation in (8.22) is written as ˜ Λ ( a ) m ( u ) = Λ ( a ) m ( u ) m Y i =1 γ a  u + m + 1 − 2 i t a  , (8.28) γ a ( u ) = η a ( u − 1 t a ) η a ( u + 1 t a ) N Y k =1 φ ( a,r k ) 1 ,s k ( u − v k ) − 1 . (8 .2 9) In view o f (8.28), replace T ( a ) m ( u ) in the o riginal T-system with T ( a ) m ( u ) Q m i =1 γ a ( u + ( m + 1 − 2 i ) /t a ). After removing the common factor , the result is indeed reduced to the form (2.22) with g ( a ) m ( u ) = m Y i =1 g ( a ) 1  u + m + 1 − 2 i t a  , (8 .30) g ( a ) 1 ( u ) = A − 1 a,z | Y c,z → γ c ( u ) ( z = q tu ) . (8.31) The prop er t y (ii) directly follo ws from (8.30) without using the concrete form of g ( a ) 1 ( u ). The prop erty (i) is essentially due to the remark after (8.22). In fact, it is attributed to g ( a ) 1 ( u ) (8.31). With regar d to γ c ( u ) therein, φ ( c,r k ) 1 ,s k ( u − v k ) in (8.29) depe nds on the quantu m space data only , and so do es the contribution from η c bec ause of (8.16) and (8.19). Remark 8 .2. The transfer ma trix (8.15) can b e gener alized b y the “magnetic ﬁeld” a s T ( a ) m ( u ) = T r W ( a ) m ( u ) ( e H R ( a,m ; r N ,s N ) 0 ,N ( z / w N ) · · · R ( a,m ; r 1 ,s 1 ) 0 , 1 ( z / w 1 )) with- out sp oiling the commutativit y a nd the T-system. Her e H is a ny element in the Cartan subalgebra of U q ( g ) acting on the auxiliary space. The dr essed v ac- uum form for suc h T ( a ) m ( u ) is obtained b y mo difying the substitution (8.17) in to Y c,q tv → e ω c ( H ) η c ( v − 1 t c ) Q c ( v − 1 t c ) η c ( v + 1 t c ) Q c ( v + 1 t c ) . Accor dingly ˜ A a,q tu (8.20) a nd the LHS of the Bethe equation (8.25) get multiplied by the extra factor e α a ( H ) . 8.3. RSOS mo dels. W e co nsider the sp ectrum of the transfer matrix T ( a ) m ( u ) (1 ≤ m ≤ t a ℓ ) (3.50) for the trigonometric level ℓ RSOS models sketc hed in Section 3 .7. ( T ( a ) t a ℓ ( u ) co rresp onds to a froz en mo del.) Conjecturally , it is covered by the dressed v acuum form in Remark 8.2 sp ecialized along (i)–(iii) in what fo llows. (i) The parameter q entering thro ugh [ u ] q t/ 2 is set q = exp  π √ − 1 t ( ℓ + h ∨ )  , where h ∨ is the dual Coxeter num b er of g (2.3). (ii) The integers n 1 , . . . , n r ent ering (8.24) ar e ﬁxed b y demanding (8.23) b e 0, which is p ossible thanks to (3 .5 1). 69 (iii) The magnetic ﬁeld is taken so that ω c ( H ) = 2 π √ − 1( ω c | Λ+ ρ ) ℓ + h ∨ , where ρ = P a ∈ I ω a and Λ is an element from P ℓ (3.45). Int ro duce the sp ecialized q -character Q ( a ) m (Λ) := χ q ( W ( a ) t a ℓ ( u )) | Y c,q tv → e ω c ( H ) , where Λ-dep endence en ters through the ab ov e H . Then according to the conjecture in [1, (A.8)-(A.9)], the r elation Q b ∈ I Q ( b ) t b ℓ (Λ) C ab = 1 holds. The qua ntit y Q ( a ) m = dim q res W ( a ) t a ℓ in Section 14.6 is equal to Q ( a ) m (0) in the no tation her e. The ab ov e relation is a generaliza tion o f Q ( a ) t a ℓ (0) = 1 in Section 14.6. 8.4. Bibliog raphical notes. The analytic Bethe ansatz was prop osed in [5 4] by extracting the idea fro m Baxter’s solution of the 8 -vertex mo de l [5 2]. It was applied systematically in [55, 144, 137] to a wide class o f solv able vertex mo dels. F ormula- tion of the B e the equatio n by ro ot system go es back, for instance, to [14 7, 55]. A relation betw e e n dres s ed v acuum for ms a nd q -characters similar to Section 8.2 has also been a rgued in [70, section 6]. 9. Wronskian type (Casora tian) formula Here we present the solutio n of the T-sy s tem for A r and C r in terms of Caso r a- tian (diﬀerence analog of W rons k ian). It is most natur ally done by intro ducing a diﬀerence a nalog o f L - op erator s in soliton theory . It also provides a Casora tia n in- terpretation and g eneraliza tio n of the Baxter Q -functions. O ur descriptio n is alo ng the context of q -characters ; hence, the identiﬁcation of the v ariables Y a,q tu = Q a ( u − 1 t a ) Q a ( u + 1 t a ) (9.1) is a ssumed. See (8.17). ( t, t a are deﬁned in (2.1).) Resulting formulas can suitably be modiﬁed to ﬁt tr a nsfer matrices with speciﬁc normalizations acco rding to the argument in Section 8.2. W e will also give analo gous L - op erator s for B r , D r and sl ( r | s ). 9.1. Diﬀerence L op erators. W e tr e at the A r case ﬁrst as an illustr ation. Let D = e 2 ∂ u be the shift op er ator D f ( u ) = f ( u + 2) D . Using z a ( u ) (7.6), we introduce the diﬀerence L op erator: L ( u ) = (1 − z r +1 ( u ) D ) · · · (1 − z 2 ( u ) D )(1 − z 1 ( u ) D ) . (9.2) Expanding the pr o duct, one identiﬁes the co eﬃcients with m = 1 ca s e of (7.4) to ﬁnd L ( u ) = r +1 X a =0 ( − 1) a T ( a ) 1 ( u + a − 1) D a , (9.3) where T (0) 1 = T ( r +1) 1 = 1. Thus L ( u ) is a gener ating function o f the fundamental q -characters T ( a ) 1 ( u ) = χ q ( W ( a ) 1 ( u )). Deﬁne the action of the screening op er ator S a (4.26) o n diﬀerence op erator s by S a · ( P i f i ( u ) D i ) = P i ( S a · f i ( u )) D i . Let us calculate S a · L ( u ) by us ing the factorized form (9.2). Accor ding to the r ule (4.26), S a acts non trivially o nly on the v ariable Y a,z . F rom (7.6), it is contained only in z a ( u ) and z a +1 ( u ). The action 70 on this part is calculated as S a · (1 − z a +1 ( u ) D )(1 − z a ( u ) D ) = S a · (1 − Y − 1 a,q u + a +1 Y a +1 ,q u + a D − Y − 1 a − 1 ,q u + a Y a,q u + a − 1 D + Y − 1 a − 1 ,q u + a +2 Y a +1 ,q u + a D 2 ) = S a,q u + a +1 Y − 1 a,q u + a +1 Y a +1 ,q u + a D − S a,q u + a − 1 Y − 1 a − 1 ,q u + a Y a,q u + a − 1 D = 0 , where the last equality is due to (4 .27) and (4.25): S a,q u + a +1 = A a,q u + a S a,q u + a − 1 = Y a,q u + a − 1 Y a,q u + a +1 Y − 1 a − 1 ,q u + a Y − 1 a +1 ,q u + a S a,q u + a − 1 . In this w ay one gets S a · L ( u ) = 0 (1 ≤ a ≤ r ) . (9.4) In view o f (9.3), this o ﬀer s a simple way of checking T ( a ) 1 ( u ) ∈ T r b =1 Ker S b in agreement with Theorem 4.6 (2). When r = 1, the change of v ariables from { z a ( u ) } to { T ( a ) 1 ( u ) } is a diﬀerence analog of the Miura tr ansformation q = q ( u ) → f = f ( u ) = q 2 − ∂ u q by ( ∂ u − q )( ∂ u + q ) = ∂ 2 u − f . With regard to the inv erse L ( u ) − 1 = (1 − z 1 ( u ) D ) − 1 (1 − z 2 ( u ) D ) − 1 · · · (1 − z r +1 ( u ) D ) − 1 , the simple expansion formula L ( u ) − 1 = X m ≥ 0 T (1) m ( u + m − 1) D m (9.5) holds due to (7.4), conﬁrming similar ly that T (1) m ( u ) ∈ T r b =1 Ker S b . The product of (9.3) and (9.5) leads to the tw o types of TT-r elations: X 0 ≤ a ≤ min( r +1 ,m ) ( − 1) a T ( a ) 1 ( u + a ) T (1) m − a ( u + m + a ) = δ m 0 , X 0 ≤ a ≤ min( r +1 ,m ) ( − 1) a T ( a ) 1 ( u + m − a ) T (1) m − a ( u − a ) = δ m 0 for m ≥ 0. 9.2. Casoratian formula. Consider the linear diﬀerence e q uation on w ( u ) L ( u ) w ( u ) = 0 . (9.6) This is of or der r + 1 with resp ect to D . Letting { w 1 ( u ) , . . . , w r +1 ( u ) } b e a bas is of the solution, we denote the Cas oratian b y C u [ i 1 , . . . , i k ] = det    w 1 ( u + i 1 ) · · · w 1 ( u + i k ) . . . . . . w k ( u + i 1 ) · · · w k ( u + i k )    (9.7) for 1 ≤ k ≤ r + 1. Thus for exa mple C u +2 [ i 1 , . . . , i k ] = C u [ i 1 + 2 , . . . , i k + 2]. By using (9.3), the rela tions L ( u ) w k ( u ) = 0 with k = 1 , . . . , r + 1 ar e ex pressed in the matrix form:      w 1 ( u ) w 2 ( u ) . . . w r +1 ( u )      =      w 1 ( u + 2) w 1 ( u + 4) · · · w 1 ( u + 2 r + 2) w 2 ( u + 2) w 2 ( u + 4) · · · w 2 ( u + 2 r + 2) . . . . . . w r +1 ( u + 2) w r +1 ( u + 4) · · · w r +1 ( u + 2 r + 2)            T (1) 1 ( u ) ( − 1) T (2) 1 ( u + 1) . . . ( − 1) r T ( r +1) 1 ( u + r )       , 71 where T ( r +1) 1 ( u ) = 1 in our normaliza tion here ( q -characters) as noted under (7 .6). By Cramer’s formula, we have T ( a ) 1 ( u + a − 1) = C u [0 , . . . , 2 a − 2 , 2 a + 2 , . . . , 2 r + 2] C u [2 , . . . , 2 r + 2] (0 ≤ a ≤ r + 1) , (9.8) where . . . sig niﬁes that the omitted arr ays ar e consecutive with diﬀerence 2 . The re- lation L ( u ) w k ( u ) = 0 means that w k ( u +2 r +2) = ( − 1) r w k ( u )+terms inv olving w k ( u + 2) , . . . , w k ( u + 2 r ). It follows the p er io dicity C u [0 , 2 , . . . , 2 r ] = C u +2 [0 , 2 , . . . , 2 r ] . (9.9) Its actual v alue b ecomes impor tant in physical a pplications, and the r e s ulting rela- tion on C u [0 , 2 , . . . , 2 r ] is ca lled the quantum W ro nskian condition. See for example [148, 149]. The solution to the T-system for A r that matches (9.8) is g iven by T ( a ) m ( u + a + m − 2) = C u [0 , . . . , 2 a − 2 , 2 a + 2 m, . . . , 2 r + 2 m ] C u [0 , . . . , 2 r ] (0 ≤ a ≤ r + 1 ) . (9.10) This satisﬁes the b oundary conditions T (0) m ( u ) = T ( a ) 0 ( u ) = 1 and T ( a ) − 1 ( u ) = 0. In fact, if (9.10) is substituted int o (2.5), the denominator can b e removed as an overall factor owing to (9.9). Then (2.5) is identiﬁed with a s imples t Pl ¨ uc ker relation ξ ( a ) m ( u ) ξ ( a ) m ( u + 2) − ξ ( a ) m +1 ( u ) ξ ( a ) m − 1 ( u + 2) − ξ ( a +1) m ( u ) ξ ( a − 1) m ( u + 2) = 0 (9 .11) among the determinant ξ ( a ) m ( u ) = C u [0 , . . . , 2 a − 2 , 2 a + 2 m, . . . , 2 r + 2 m ]. The Caso ratian formula (9.10) is a Y ang-Ba xterization ( u -dep endent general- ization) of the W eyl character formula. T o see this, recall the r estriction map res (4.23). F rom (7.6) we hav e r es ( z a ( u )) = x a , where the latter is deﬁned by x a = y a /y a − 1 = e ω a − ω a − 1 with ω 0 = ω r +1 = 0. W e extend res naturally to the diﬀerence L op e rator and the wa ve functions as res L ( u ) = (1 − x r +1 D ) · · · (1 − x 1 D ) , res ( w i ( u )) = x − u/ 2 i . (9.1 2) The latter is certainly annihilated by the former. By using x 1 · · · x r +1 = 1, it is straightforward to see that the r estriction of (9.10) b ecomes res  C u [0 , . . . , 2 a − 2 , 2 a + 2 m, . . . , 2 r + 2 m ] C u [0 , . . . , 2 r ]  = det( x λ j + r +1 − j i ) 1 ≤ i,j ≤ r +1 det( x r +1 − j i ) 1 ≤ i,j ≤ r +1 , (9.13) where ( λ j ) cor resp onds to the a × m re ctangular Y oung diagra m, namely , λ j = m if 1 ≤ j ≤ a and λ j = 0 o therwise. The RHS is the W eyl character formula of the Sch ur function for ( λ j ) as is well known. The Cas oratian formula her e and the tableau sum formula (Section 7.1) ar e connected b y the following gener al fact. Prop ositio n 9.1 ([142]) . L et C u [ i 1 , . . . , i k ] b e as in (9.7). ( L ( u ) w j ( u ) = 0 is not assume d.) Given even int e gers 0 = i 0 < i 1 < · · · < i N − 1 , let µ = ( µ j ) b e the Y oung diagr am with depth less than N sp e ciﬁe d by µ j = i N − j 2 + j − N . T ake any d ≥ µ 1 . Then C u [0 , i 1 , i 2 , · · · , i N − 1 ] C u +2 d [0 , 2 , . . . , 2 N − 2] = X T Y ( α,β ) ∈ ( d N ) /µ ˜ x T ( α,β ) ( u + 2 α + 2 β − 4) , 72 wher e ˜ x j ( u ) = C u [0 , 2 ,..., 2 j − 2] C u [4 , 6 ,..., 2 j ] C u [2 , 4 ,..., 2 j ] C u [2 , 4 ,..., 2 j − 2] and the sum P T extends over the semis- tandar d table aux on the skew Y oung diagr am ( d N ) /µ [135] on letters { 1 , . . . , N } . T ( α, β ) denotes t he entry of T at t he α th r ow and t he β th c olumn fr om the b ott om left c orner. According to Prop ositio n 9.1, the RHS of (9.10) equals the sum ov er semistan- dard tableaux on a × m Y oung diagra m on letters { 1 , . . . , r + 1 } . The building blo ck of the tableau v aria ble ˜ x j ( u ) is the principal mino rs of the Casora tian (quan- tum W ronskia n) C u [0 , 2 , . . . , 2 r ]. Co m bined with (9.6), they are identiﬁed w ith the Baxter Q -functions as we will see in the next s ubsection. 9.3. Q -functions. F r om the full L op era tor (9.2), we extract the par tial ones b y L j ( u ) = (1 − z j ( u ) D ) · · · (1 − z 2 ( u ) D )(1 − z 1 ( u ) D ) (1 ≤ j ≤ r + 1) . (9.14 ) The original one corr esp onds to L r +1 ( u ). By the deﬁnition we hav e Ker L 1 ( u ) ⊂ K er L 2 ( u ) ⊂ · · · ⊂ Ker L r +1 ( u ) . (9.15 ) Cho ose the basis of Ker L j ( u ) accor ding to this ﬂag s tructure as { w 1 ( u ) } ⊂ { w 1 ( u ) , w 2 ( u ) } ⊂ · · · ⊂ { w 1 ( u ) , . . . , w r +1 ( u ) } . (9.16) As the simplest example, w 1 ( u ) ∈ Ker L 1 ( u ) is the condition 0 = (1 − z 1 ( u ) D ) w 1 ( u ). In view of (7.6) and (9.7), this is the j = 1 case of  1 − Y j,q u + j − 1 D  C u [0 , . . . , 2 j − 2] = 0 (1 ≤ j ≤ r ) . (9.17) T o derive this, note tha t a direct calculation using (7.6) leads to L j ( u ) = 1 + ( − 1) j Y j,q u + j − 1 D j + terms inv o lving D , . . . , D j − 1 . Therefore L j ( u ) w k ( u ) = 0 (1 ≤ k ≤ j ) implies Y j,q u + j − 1 w k ( u + 2 j ) = ( − 1) j − 1 w k ( u ) + j − 1 X l =1 c j,l ( u ) w k ( u + 2 l ) , where c j,l ( u ) is indepe ndent of k . T he second term in (9.17) is equa l to Y j,q u + j − 1 C u [2 , . . . , 2 j − 2 , 2 j ]. Applying the ab ov e relation to the last co lumn of this, we ﬁnd the res ult is equal to C u [0 , . . . , 2 j − 2], hence (9.17). If we express the v aria ble Y a,q u in q -character s in terms o f Q -functions as in (9.1), the solution of the ﬁrst order diﬀerence equa tion (9.17) is given by C u [0 , . . . , 2 j − 2] = σ j ( u ) Q j ( u + j − 2) (1 ≤ j ≤ r ) , (9.18 ) where σ j ( u ) is any v ariable satisfying σ j ( u + 2) = σ j ( u ). In this wa y , the Q -functions are identiﬁed with the principa l minors of the Cas oratian C u [0 , . . . , 2 r ] u made of the wa ve functions { w i ( u ) } esp ecially chosen along the scheme (9.16). The simplest case j = 1 o f (9.18) is w 1 ( u ) = σ 1 ( u ) Q 1 ( u − 1). Thus L ( u ) w 1 ( u ) = 0 is r ephrased as r +1 X a =0 ( − 1) a T ( a ) 1 ( u + a ) Q 1 ( u + 2 a ) = 0 , (9.19) which is an example of TQ- relations. 73 9.4. B¨ ac klund transformations. Here we r emov e the boundar y co ndition T ( a ) 0 ( u ) = T (0) m ( u ) = 1 a nd redeﬁne T ( a ) m ( u ) in (9.10) and Q j ( u ) in (9.18) as T ( a ) m ( u + a + m − 2) = C u [0 , . . . , 2 a − 2 , 2 a + 2 m, . . . , 2 r + 2 m ] , (9.2 0 ) Q a ( u + a − 1) = C u [0 , . . . , 2 a − 2] . (9.21) These functions are spe cial cases of more gene r al ones: T ( s,a ) m ( u + a + m − 2) =        w 1 ( u ) · · · w 1 ( u + 2 a − 2) w 1 ( u + 2 a + 2 m ) · · · w 1 ( u + 2 s + 2 m ) . . . . . . w s +1 ( u ) · · · w s +1 ( u + 2 a − 2) w s +1 ( u + 2 a + 2 m ) · · · w s +1 ( u + 2 s + 2 m )        , Q { i 1 ,...,i a } ( u + a − 1) =        w i 1 ( u ) · · · w i 1 ( u + 2 a − 2) . . . . . . w i a ( u ) · · · w i a ( u + 2 a − 2)        , (9.22) where · · · in determinants signify that u incr eases b y 2. T ( s,a ) m ( u ) is deﬁned for 0 ≤ a ≤ s + 1 , 0 ≤ s ≤ r and m ≥ 0. The set { i 1 , . . . , i a } is any subset o f { 1 , . . . , r + 1 } . By the deﬁnition, T ( r,a ) m ( u ) = T ( a ) m ( u ) and Q { 1 ,...,a } ( u ) = Q a ( u ). Thes e functions ob ey v arious relatio ns a s the conse q uence o f identit ies a mong determinants. L e t us men tion a few of them that hav e analogy with soliton theory . The s y mmetric gr o up S r +1 acts on the ba sis w 1 ( u ) , . . . , w r +1 ( u ) as their per mu- tations keeping L ( u ) inv ariant. This can be view ed as B¨ acklund tra nsformations generating the functions Q { i 1 ,...,i a } from Q 1 , . . . , Q r +1 . Its gener ator, the transp os i- tion s a of w a ( u ) and w a +1 ( u ), acts triv ia lly as s a ( Q b ) = Q b for a > b and similarly as s a ( Q b ) = − Q b for a < b . The nontrivial ca se s a ( Q a ) = Q { 1 ,...,a − 1 ,a +1 } satisﬁes the QQ-re lation: D ( Q a ) s a ( Q a ) − Q a D s a ( Q a ) + D ( Q a − 1 ) Q a +1 = 0 , (9.23) where the ﬁrst term denotes Q a ( u + 2) s a ( Q a )( u ) for instance. This is derived by applying the Jacobi iden tity (6 .2) to the a, a + 1 r ows and 1 , a + 1 columns for the determinant of Q a +1 . With regar d to T ( s,a ) m ( u ), it is the T- function for A s ( ⊂ A r ). W r iting T ( s,a ) m ( u ) and T ( s − 1 ,a ) m ( u ) simply as T ( a ) m ( u ) and ˜ T ( a ) m ( u ), resp ectively , one can derive T ( a ) m ( u ) ˜ T ( a − 1) m ( u − 1) = T ( a − 1) m ( u − 1) ˜ T ( a ) m ( u ) + T ( a ) m − 1 ( u − 1) ˜ T ( a − 1) m +1 ( u ) , T ( a ) m +1 ( u − 1) ˜ T ( a ) m ( u ) = T ( a ) m ( u ) ˜ T ( a ) m +1 ( u ) + T ( a +1) m ( u − 1) ˜ T ( a − 1) m +1 ( u ) (9.24) from the Pl ¨ uc ker r elation. This is a B¨ acklund trans fo rmation betw een T-functions asso ciated with A s and A s − 1 . The T-system for T ( a ) m ( u ) ar ises a s a compatibility of the tw o linea r equations on ˜ T ( a ) m ( u ) [150]. F or mor e examples, see [151, 15 2, 153, 22, 23] and r eferences therein. It is a n op en pr oblem to construct such a Lax representation of the T-sy stem for general g . 9.5. T yp e C r . Let D b e the diﬀerence op er ator D f ( u ) = f ( u + 1) D . W e use the v ariable z a ( u ) ( a ∈ J ) (7.17) which ar e r elated to the Q -functions by (9.1 ). W e also 74 int ro duce the v ariables x 1 ( u ) , . . . , x 2 r + 2 ( u ) by x a ( u ) = z a ( u ) , x 2 r + 3 − a ( u ) = z a ( u ) (1 ≤ a ≤ r ) , x r +1 ( u ) = − x r +2 ( u ) = Q r ( u + r − 1 2 ) Q r ( u + r +3 2 ) Q r ( u + r +1 2 ) 2 . (9.25) Note that x r +1 ( u ) a nd x r +2 ( u ) a re n ot contained in Z [ Y ± a,z ] a ∈ I ,z ∈ C × . With the notation − → Y 1 ≤ i ≤ k X i = X 1 X 2 · · · X k , ← − Y 1 ≤ i ≤ k X i = X k · · · X 2 X 1 , (9.26 ) the diﬀerence L -op era tor is L ( u ) = − → Y 1 ≤ a ≤ r (1 − z a ( u ) D ) · (1 − z r ( u ) z r ( u + 1) D 2 ) · ← − Y 1 ≤ a ≤ r (1 − z a ( u ) D ) . (9.27) One ca n easily ch eck S a · L ( u ) = 0 as in t yp e A . The middle quadra tic op er a tor can be factorized as 1 − Y r,q 2 u + r +1 Y − 1 r,q 2 u + r +3 D 2 = 1 − Q r ( u + r +5 2 ) Q r ( u + r − 1 2 ) Q r ( u + r +1 2 ) Q r ( u + r +3 2 ) D 2 = (1 ± x r +2 ( u ) D )(1 ± x r +1 ( u ) D ) . Thu s (9.27) is expressed as L ( u ) = ← − Y 1 ≤ i ≤ 2 r +2 (1 − x i ( u ) D ) , (9.28) which resembles cur iously the A 2 r + 1 case rather tha n A 2 r − 1 . The op era tor L ( u ) generates each fundamental q -character “twice”. Theorem 9. 2 ([13 9]) . L ( u ) = r X a =0 ( − 1) T ( a ) 1 ( u + a − 1 2 ) D a − 2 r + 2 X a = r +2 ( − 1) a T (2 r +2 − a ) 1 ( u + a − 1 2 ) D a , wher e T (0) 1 = 1 . F rom Theor em 9.2 a nd (9.28), w e obtain a no ther table a u sum formula for the fundamen tal q -characters: T ( a ) 1 ( u + a − 1 2 ) = X 1 ≤ i 1 ≤···≤ i a ≤ 2 r +2 a Y k =1 x i k ( u + a − k ) (1 ≤ a ≤ r ) . (9.29) Although this is forma lly the same form as A 2 r + 1 case (7.4 ), the v a riable x r +2 ( u ) (9.25) is “negative” here. It is hig hly nontrivial that the cancellation due to the s ign yields the previous form ula (7.21) describ e d by the rule (7.20), which cons titutes a substantial par t of the pr o of of Theorem 9.2. O n the other hand it is easy to see L ( u ) − 1 = X m ≥ 0 T (1) m ( u + m − 1 2 ) D m (9.30) from (7.19), (7.18) and (9.27). 75 The rest of this subsection will b e brief as the con tent is more o r less pa rallel with A 2 r + 1 case. W e formally extend the fundamental q - character s T ( a ) 1 ( u ) to 1 ≤ a ≤ 2 r + 2 by T ( a ) 1 ( u ) + T (2 r +2 − a ) 1 ( u ) = 0 (0 ≤ a ≤ 2 r + 2) . (9.31) Then Theorem 9.2 is rephrased as L ( u ) = 2 r + 2 X a =0 ( − 1) a T ( a ) 1 ( u + a − 1 2 ) D a . (9.32) W e consider the diﬀerence equation L ( u ) w ( u ) = 0 and a basis of the solutio n { w 1 ( u ) , . . . , w 2 r + 2 ( u ) } . With the same notation C u [ i 1 , . . . , i k ] as (9.7), we hav e the Casora tian formula T ( a ) 1 ( u + a − 1 2 ) = C u [0 , . . . , a − 1 , a + 1 , . . . , 2 r + 2] C u [1 , . . . , 2 r + 2] (0 ≤ a ≤ 2 r + 2) , (9.33) where . . . signiﬁes that the omitted arrays a re consecutive with diﬀerence 1 . The denominator po ssesses the p erio dicity C u [0 , 1 , . . . , 2 r + 1] = − C u +1 [0 , 1 , . . . , 2 r + 1] , (9.34) which is a C r analog of the quantum W r onskian condition. Set ξ ( a ) m ( u ) = C u [0 , . . . , a − 1 , a + m, . . . , 2 r + 1 + m ] , ξ ( u ) = C u [0 , 1 , . . . , 2 r + 1] . (9.35) The solution of the unr estricted T-system for C r that matches (9.33) is g iven by Theorem 9. 3 ([13 9]) . The fol lowing is a solut ion of the T-syst em for C r . T ( a ) m ( u + a + m − 2 2 ) = ( − 1 ) m − 1 ξ ( a ) m ( u ) ξ ( u + 1) (1 ≤ a ≤ r − 1) , T ( r ) m ( u + r + 2 m − 1 2 ) T ( r ) m ( u + r + 2 m − 3 2 ) = ξ ( r ) 2 m ( u ) ξ ( u ) , T ( r ) m ( u + r + 2 m − 1 2 ) T ( r ) m +1 ( u + r + 2 m − 1 2 ) = ξ ( r ) 2 m +1 ( u ) ξ ( u + 1) , T ( r ) m ( u + r + 2 m − 1 2 ) 2 = ξ ( r +1) 2 m ( u ) ξ ( u ) . As for the ﬁrst three, there is an alternative expressio n der ived b y using the ident it y ξ ( a ) m ( u ) = ( − 1) a + m + r +1 ξ (2 r +2 − a ) m ( u + a − r − 1). See Prop osition 4 .3 in [139] for details. 9.6. T yp e B r and D r . Here we only give the L -op era tors and their expansio ns. Let D b e the diﬀerence op erator D f ( u ) = f ( u + 2) D . W e use the v aria bles z a ( u ) for B r (7.9) and D r (7.23) which ar e rela ted to the Q -function b y (9 .1). The diﬀer ence 76 L -op era tors are B r : L ( u ) = − → Y 1 ≤ a ≤ r (1 − z a ( u ) D ) · (1 + z 0 ( u ) D ) − 1 · ← − Y 1 ≤ a ≤ r (1 − z a ( u ) D ) , (9.36 ) D r : L ( u ) = − → Y 1 ≤ a ≤ r (1 − z a ( u ) D ) · (1 − z r ( u ) z r ( u + 2) D 2 ) − 1 · ← − Y 1 ≤ a ≤ r (1 − z a ( u ) D ) . (9.37) One can c heck S a · L ( u ) = 0 by ex pa nding the middle factor into a p ow er ser ies in D . Introduce the expa nsion c o eﬃcients o f L ( u ) as L ( u ) = X a ≥ 0 ( − 1) a T a ( u + a − 1) D a , L ( u ) − 1 = X m ≥ 0 T m ( u + m − 1) D m . (9 .38) They are related to the previous tableau cons tr uctions as follo ws: T m ( u ) = T (1) m ( u ) (7.12) for B r and (7.25) for D r , T a ( u ) = T ( a ) t a ( u ) (7.12) for B r , 1 ≤ a ≤ r and (7.27) for D r , 1 ≤ a ≤ r − 2 . With the co nv ention T a ( u ) = 0 for a < 0, the co eﬃcient T a ( u ) b eyond these upper bo und is characterized b y the following relations with the q - characters of spin representations: B r : T a ( u ) + T h ∨ − a ( u ) = T ( r ) 1 ( u + h ∨ 2 − a ) T ( r ) 1 ( u − h ∨ 2 + a ) , (9.3 9) D r : T a ( u ) + T h ∨ − a ( u ) = T ( r ) 1 ( u + h ∨ 2 − a ) T ( r − δ ) 1 ( u − h ∨ 2 + a ) + T ( r − 1) 1 ( u + h ∨ 2 − a ) T ( r − 1+ δ ) 1 ( u − h ∨ 2 + a ) . (9.40) Here a ∈ Z is a rbitrary and δ = 0 if a ≡ r mo d 2 and δ = 1 other wise. h ∨ is the dual Coxeter num b er (2.3), i.e. h ∨ = 2 r − 1 for B r and h ∨ = 2 r − 2 for D r . In particular, one has T r − 1 ( u ) = T ( r ) 1 ( u ) T ( r − 1) 1 ( u ) for D r . 9.7. T yp e sl ( r | s ). There are t wo kinds of ro ots, o dd and even for the g raded algebra sl ( r | s ). The choice of simple ro ots is not unique. The most standar d one is called distinguished, where all r o ots but α r is even. Here we follow [19] and set I = { 1 , · · · , r + s } = I 1 ∪ I 2 , I 1 = { 1 , 2 , . . . , r } , I 2 = { r + 1 , r + 2 , . . . , r + s } , and assign the grading p a by p a = 1 ( − 1) if a ∈ I 1 (I 2 ). The Cartan matrix is ex pressed by the gra ding a s ( α k | α j ) = ( p k + p k +1 ) δ kj − p k +1 δ k +1 ,j − p k δ k,j +1 . Now the analo g o f (7.6) is z a ( u ) = Y − p a a − 1 ,q u + s a Y p a a,q u + s a − 1 ( a ∈ I) , 77 where s a = P a j =1 p j and Y 0 ,q u = Y r + s,q u = 1. Let D be the diﬀerence ope r ator D f ( u ) = f ( u + 2) D . Then the analog of (9.3) and (9 .5) a r e given as (1 + z r + s ( u ) D ) p r + s · · · (1 + z 1 ( u ) D ) p 1 = ∞ X a =0 T ( a ) 1 ( u + a − 1) D a , (1 − z 1 ( u ) D ) − p 1 · · · (1 − z r + s ( u ) D ) − p r + s = ∞ X m =0 T (1) m ( u + m − 1) D m . Example 9 .4. sl (2 | 1), p 1 = p 2 = − p 3 = 1. T (1) 1 ( u ) = Y 1 ,z + Y − 1 1 ,z q 2 Y 2 ,z q − Y 2 ,z q , T (2) 1 ( u ) = Y 2 ,z − Y 1 ,z q Y 2 ,z − Y − 1 1 ,z q 3 Y 2 ,z Y 2 ,z q 2 + Y 2 ,z Y 2 ,z q 2 , T (3) 1 ( u ) = − Y 2 ,z q − 1 Y 2 ,z q + Y 1 ,z q 2 Y 2 ,z q − 1 Y 2 ,z q + Y − 1 1 ,z q 4 Y 2 ,z q − 1 Y 2 ,z q Y 2 ,z q 3 − Y 2 ,z q − 1 Y 2 ,z q Y 2 ,z q 3 . sl (2 | 1), p 1 = − p 2 = p 3 = 1. T (1) 1 ( u ) = Y 1 ,z − Y 1 ,z Y − 1 2 ,z q + Y − 1 2 ,z q , T (2) 1 ( u ) = − Y 1 ,z q − 1 Y 1 ,z q Y − 1 2 ,z + Y 1 ,z q Y − 1 2 ,z + Y 1 ,z q − 1 Y 1 ,z q Y − 1 2 ,z q 2 Y − 1 2 ,z − Y 1 ,z q Y − 1 2 ,z q 2 Y − 1 2 ,z , T (3) 1 ( u ) = Y 1 ,z q − 2 Y 1 ,z Y 1 ,z q 2 Y − 1 2 ,z q − 1 Y − 1 2 ,z q − Y 1 ,z Y 1 ,z q 2 Y − 1 2 ,z q − 1 Y − 1 2 ,z q , − Y 1 ,z q − 2 Y 1 ,z Y 1 ,z q 2 Y − 1 2 ,z q − 1 Y − 1 2 ,z q Y − 1 2 ,z q 3 + Y 1 ,z Y 1 ,z q 2 Y − 1 2 ,z q − 1 Y − 1 2 ,z q Y − 1 2 ,z q 3 . F or the formulas for general case, see [152, 153]. 9.8. Bibliog raphical notes. The Casoratia n solutio n (9.10) for A r has b een known in v arious contexts. F or the T-system of tra ns fer matrices, a s lightly mo re general solution than (9.20) was g iven in eq.(2.25 ) in [150] containing 2 r + 2 a rbitrary func- tions. It does not satisfy the natural b ounda ry condition T ( a ) − 1 ( u ) = 0 for fusion transfer matrices in gener al. As usual, such a “Dirichlet” condition halves the arbi- trary functions to w 1 ( u ) , . . . , w r +1 ( u ), whic h brings one back to (9 .20). Casor atian solutions are known also for the r estricted T-systems for A r [124] and C r [17]. The L -op era tor for t yp e A has b een studied from the viewp oint of diﬀerence analog of Drinfeld-Sokolov reduction [154]. The co ncrete fo rms for type B C D and their application to q -characters were given in [139]. Analog ous diﬀer ence L - op erator s for a ll the twisted case s except E (2) 6 hav e been constr uc ted in [155]. The results (9.39) and (9.40) ar e taken from Theor em 2.3 in [13 7] a nd Pr op osition 2 .3 in [138], resp ectively . 10. T-system in ODE T-system a ppea rs also in the connec tion problem of 1 D Schr¨ odinger e quation, which is a typical exa mple of the ODE (ordinar y diﬀerential equa tions)/IM (inte- grable mo dels ) corres p o ndence. As a comprehensible review on the ODE/IM cor - resp ondence is a lready av ailable in [14 9], we o nly discuss the issue br ieﬂy in view of T- system. W ronskians appear naturally in the cont ext of ODE. They will b e shown to coincide with the analogous ob ject, the Ca soratia n (9.7) in the diﬀerence equation in Section 9. 78 10.1. Generalized Stokes multipliers - the 2nd order case. As the simplest example, we consider the 1 D Sc hr¨ odinger equation on the r eal ax is with a p otential term:  − d 2 dx 2 + x 2 M  ψ ( x ) = E ψ ( x ) , (10.1) where M ∈ Z > 0 . The b oundary co ndition ψ ( ± ∞ ) = 0 is impo sed. W e ﬁnd it conv enient to extend x into the complex plane 20 . Since the Schr¨ odinger eq uation has the irregula r singular ity at ∞ , we ex pec t a sudden change o f ψ ( x ) when cro ssing a b o r der line of sectors deﬁned b elow. This is called the Sto kes phenomenon. The ch ange is characteriz e d by the Stokes multiplier τ 1 . Belo w we will introduce a s e t of genera lized Stokes multipliers { τ j } 2 M j =1 and show that they satisfy the level 2 M restricted T-system for A 1 . First, let S j be a sector in the complex plane deﬁned by S j = n x      arg x − j π M + 1    < π 2 M + 2 o . The sector S 0 th us includes the p os itive rea l axis . W e then in tro duce a solution φ ( x, E ) to (10 .1) which decays exp onentially as x tends to ∞ inside S 0 as φ ( x, E ) ∼ x − M / 2 √ 2 i exp  − x M +1 M + 1  , x ∈ S 0 . (10.2) This is refer red to as the sub dominant solution. There should b e another solution to (10 .1) which diverges ex po nentially in S 0 as x tends to ∞ . W e call it dominant. It is also r epresented by φ . T o see this, note the in v ariance of (10.1) under the simult aneous tra nsformations x → q − 1 x a nd E → E q 2 , where q = exp( π i M +1 ). W e call this “discrete rotationa l symmetr y ”. W e thus in tro duce y j = q j / 2 φ ( q − j x, q 2 j E ) so that y 0 = φ . The ab ove observ ation tells tha t any y j is a solution to (10.1). Moreov er, we can sho w that the pair ( y j , y j +1 ) forms the fundamental sys tem of solutions (FSS) in S j . This is eas ily see n by intro ducing the W ronsk ian ma tr ix Φ j and the W ronskia n W [ y i , y j ]: Φ j =  y j y j +1 ∂ y j ∂ y j +1  , W [ y i , y j ] = det  y i y j ∂ y i ∂ y j  . By using the asymptotic form (10.2), o ne ca n chec k W [ y j , y j +1 ] = 1 , hence the pair ( y j , y j +1 ) is independent. Th us , y 0 (equals to φ ) is the sub dominant solutio n in S 0 , while y 1 is a dominant one. W e are interested in the relation among FSS in diﬀerent sec tors. Let us start from S 0 and S 1 . Obviously y 2 m ust b e represented by the linear combination o f y 0 and y 1 as y 2 = a 0 y 0 + a 1 y 1 . As W [ y j , y j +1 ] = 1 for a n y j , we ﬁnd a 0 = − 1. The co eﬃcient a 1 can b e r egarded as a function of E and w e write it as τ 1 ( E ) = a 1 , which is referred to as the Stokes m ultiplier. The result ca n b e nea tly represented in the matrix form Φ 0 = Φ 1 M 1 , 0 , M 1 , 0 =  τ 1 ( E ) 1 − 1 0  . The gener al adjacent FSS Φ j and Φ j +1 are connec ted by Φ j = Φ j +1 M j +1 ,j , and the “dis crete ro tational s ymmetry” lea ds to M j +1 ,j = M 1 , 0 | E → E q 2 j . W e in tro duce 20 F or a general reference to ODE in the complex domain, we recommend [ 156]. 79 the matrix connecting well sepa rated sectors Φ 0 = Φ j M j, 0 . (10.3) By the deﬁnition, the recurs ion r elation M j, 0 = M j, 1 M 1 , 0 (10.4) holds. The solution to this takes the form M j, 0 =  τ j ( E ) τ j − 1 ( E q 2 ) − τ j − 1 ( E ) − τ j − 2 ( E q 2 )  . (10.5) Here τ j is the function uniquely determined fro m τ 1 and the recur s ion r elation τ j ( q 2 E ) τ 1 ( E ) = τ j +1 ( E ) + τ j − 1 ( q 4 E ) (10.6) with τ 0 ( E ) = 1 . W e set τ − 1 ( E ) = 0 so tha t this ho lds also at j = 0. In addition we hav e τ 2 M ( E ) = 1 , τ 2 M +1 ( E ) = 0 as after 36 0 ◦ rotation, FSS must come coinc ide with the o riginal one times ( − 1). (cf. [156, (21.31 )].) W e call τ j ( j ≥ 2) gener alized Stokes mult ipliers. The generalize d Stokes m ultipliers sa tisfy the relation τ j ( E ) τ j ( E q 2 ) = τ j − 1 ( E q 2 ) τ j +1 ( E ) + 1 . (10.7) This is equiv alent to det M j, 0 = 1. It is shown either b y (10.3) or by induction on j us ing (10.6). See also the dis cussion in Section 10.3. Setting T j ( u ) = τ j ( E q − j − 1 ) , where E = exp  π iu M + 1  , we therefore have Prop ositio n 10.1. { T j ( u ) } satisfy the level 2 M r est ricte d T-s ystem for A 1 T j ( u + 1) T j ( u − 1) = T j − 1 ( u ) T j +1 ( u ) + 1 ( j = 1 , · · · , 2 M ) , (10.8) wher e T 0 ( u ) = 1 and T 2 M +1 ( u ) = 0 . Example 1 0.2. By (10.3), (10.5) and det M j, 0 = 1, one has τ j ( E ) = W [ y 0 , y j +1 ] , where the RHS is indep endent o f x . The consistency of τ 2 M = 1 and τ 2 M +1 = 0 with y 2 M +1 = − y − 1 and y 2 M +2 = − y 0 is reconﬁr med. Relation (10.7) is also re-derived from the simple identit y among W ronskians [ y α , y β ][ y γ , y δ ] = [ y α , y γ ][ y β , y δ ]+[ y α , y δ ][ y γ , y β ] by the sp ecializatio n α = 0 , β = j + 1 , γ = 1 , δ = j + 2 . No te W [ y k , y k +1 ] = 1 for any k . 10.2. Higher order ODE. O ne can e xtend the observ ation on the second o rder ODE to higher order case corre sp o nding to g = A r [157, 1 58, 159, 1 60]. Consider a natural generaliz a tion of (10 .1): ( − 1) r d r +1 y dx r +1 + x ℓ y = E y = λ r +1 y . (10.9) Let q = e iθ with θ = 2 π ℓ + r +1 . The sector S k is now deﬁned by | ar g x − k θ | ≤ θ 2 . W e pay attention to the solution φ ( x, λ ) in S 0 which decays most rapidly a s x → ∞ a s φ ( x, λ ) ∼ C x − r ℓ/ (2 r +2) exp  − x ν ν  , ν = ℓ + r + 1 r + 1 . The no rmalization factor C will b e determined later. As in the 2nd order O DE case, (1 0 .9) is in v aria nt under x → xq − 1 , E → E q r +1 . Thus in terms o f λ , y k = q r k/ 2 φ ( xq − k , λq k ) is also a solutio n to (10.9) for any k ∈ Z . 80 The FSS in S k consists of ( y k , · · · , y k + r ). It is conv e nie nt to introduce a W r on- skian matrix Φ k =    y k y k +1 · · · y k + r . . . . . . ∂ r y k ∂ r y k +1 · · · ∂ r y k + r    . W e write the determinant o f a slightly more gene r al matrix (for m ≤ r ) a s W [ y i 0 , y i 1 , · · · , y i m ] = det    y i 0 y i 1 · · · y i m . . . . . . ∂ m y i 0 ∂ m y i 1 · · · ∂ m y i m    . (10.10) Due to (10.9), the W ronskia ns ( m = r cases) ar e indep endent of x . In particular, the no rmalization constant C can be ﬁxed so that det Φ k = W [ y k , · · · , y k + r ] = 1 for any k . W e introduce the connection ma trix M k +1 ,k by Φ k = Φ k +1 M k +1 ,k . ( 10.11) It has the form M k +1 ,k =         τ (1) 1 ( λq k ) 1 0 0 · · · 0 τ (2) 1 ( λq k ) 0 1 0 · · · 0 . . . . . . τ ( r ) 1 ( λq k ) 0 0 0 · · · 1 τ ( r +1) 1 ( λq k ) 0 0 0 · · · 0         . By using Cramer’s formula, τ ( a ) 1 ( λq k ) is express ed as the W ronskian τ ( a ) 1 ( λq k ) = W [ y k +1 , · · · , y k + a − 1 , y k , y k + a +1 , · · · , y k + r +1 ] . Esp ecially , one ﬁnds τ ( r +1) 1 ( λq k ) = ( − 1) r . W e further in tro duce the generalized Stokes mult ipliers τ ( a ) m ( λ ) for m ≥ 2 by τ ( a ) m ( λ ) = W [ y 1 , y 2 , · · · y a − 1 , y 0 , y a + m , y a + m +1 · · · y r + m ] . (10.12) Note that m do es not extend to inﬁnity . Due to y r +1+ ℓ = ( − ) r y 0 , one has τ ( a ) ℓ +1 ( λ ) = 0. This causes a truncation analogo us to the lev el restriction in quantum group at ro ot of unit y . It is elementary to prov e Prop ositio n 10.3. The gener alize d Stokes mu lt ipliers τ ( a ) m ( λ ) satisfy the level ℓ r estricte d T-syst em for A r τ ( a ) m ( λ ) τ ( a ) m ( λq ) = τ ( a ) m +1 ( λ ) τ ( a ) m − 1 ( λq ) + τ ( a +1) m ( λ ) τ ( a − 1) m ( λq ) (1 ≤ a ≤ r ) , wher e the b oundary c onditions ar e m o diﬁe d as τ (0) m ( λ ) = 1 , τ ( r +1) m ( λ ) = ( − 1) r and τ ( a ) 0 ( λ ) = ( − 1) a − 1 . Remark 10. 4. One migh t exp ect that τ ( a ) m ( λ ) may app ear in the g eneralized con- nection matrix M k + m,k connecting Φ k and Φ k + m ( m ≥ 2 ). This is not the cas e. As the Sch ur functions , one can deﬁne generalized Stokes mult ipliers a sso ciated with (skew) Y oung tableaux of a ge neral shap e. Entries o f M k + m,k are genera lly identi- ﬁed with such ob jects. E sp ecially the ( a, 1 ) comp onent of M k + m,k corres p o nds to the Y oung tableau of the ho o k sha pe of width m and he ig ht a . 81 10.3. W ronskian-Casoratian duali t y . The ( i + 1 , 1 ) element from the matrix relation (10.11) with k = 0 rea ds ∂ i y 0 = τ (1) 1 ( λ ) ∂ i y 1 + · · · + τ ( r +1) 1 ( λ ) ∂ i y r +1 . Re- mem ber that y k = q r k/ 2 φ ( xq − k , λq k ) in volves x but τ ( a ) 1 ( λ ) does not. Thus one obtains an x -independent rela tion by setting x = 0 as ∂ i y 0 | x =0 = τ (1) 1 ( λ ) ∂ i y 1 | x =0 + · · · + τ ( r +1) 1 ( λ ) ∂ i y r +1 | x =0 (0 ≤ i ≤ r ) . (10.13 ) In view of y k = q r k/ 2 φ ( xq − k , λq k ), this has the sa me for m as the diﬀerence equation (TQ-relatio n) (9.6) with (9.3): w ( u ) − T (1) 1 ( u ) w ( u + 2) + · · · + ( − 1 ) r +1 T ( r +1) 1 ( u + r ) w ( u + 2 r + 2) = 0 . (10.14) In fact, under the forma l (ODE/ IM) c o rresp ondence b etw een the Stokes multipliers and the transfer matrix eigenv alues τ ( a ) 1 ( λ ) = ( − 1) a − 1 T ( a ) 1 ( u + a − 1) (1 ≤ a ≤ r + 1) , (10.15 ) the ident iﬁcation w ( u + 2 j ) = ∂ i y j | x =0 provides a so lution to (10.14) for any 0 ≤ i ≤ r . The v ariables u and λ are rela ted so that the shift u → u + 2 corresp onds to λ → λq . Now we a re entit led to substitute w i ( u + 2 j ) = ∂ i − 1 y j | x =0 (1 ≤ i ≤ r + 1) ( 10.16) int o the Casora tian C u (9.7). The result is the eq ua lity W [ y i 1 , . . . , y i k ] | x =0 = C u [2 i 1 , . . . , 2 i k ] , (10.17) which we call the W ro nskian-Cas oratian duality . One can remove “ | x =0 ” when k = r + 1 . Remember that in Section 9 .1 – 9 .3, a v ar iet y of genera lizations of T ( a ) 1 are express ed in terms of Casoratia ns C u . The re lations (10.15) and (10.17) enable us to imp ort those re sults to establis h a num b er of W ro nskian formulas for the generalized Stokes multipliers. F or example, the formula (9.10) leads to (10.12). The W ronsk ia n-Casor atian duality further provides the Stokes multipliers with dressed v a c uum forms like the o nes for A r in Section 8. Recall tha t Prop os ition 9.1 expresses the Ca soratia ns as the sums ov er semistandard tableaux like (skew) Sch ur functions. The v ariables attached to tablea u letters a re r atio of the principa l minors of C u [0 , 2 , . . . , 2 r ], namely Q a ( u + a − 1) = C u [0 , . . . , 2 a − 2] (9.21), which ar e called Baxter’s Q-functions. Via the W ronskia n- Casora tian dualit y , this is transla ted to a dressed v acuum form for Stokes mult ipliers. The tableau v ariables a re ratio of W [ y k +1 , y k +2 , . . . , y k + a ] | x =0 , which are to b e identiﬁed with Baxter ’s Q- functions Q a ( λq a + k ) in the present co nt ext. As explained in Section 9.4 for Casor atians, the solutions w 1 , . . . , w r +1 to (10.14) may b e r enum b ered arbitr arily , a nd this freedom gener ates B ¨ acklund transforma - tions among Q-functions. Even mor e genera lly , one may consider arbitrary linear combinations of (10.1 3) ins tea d of (10.16) as w i ( u + 2 j ) = r X n =0 A in ∂ n y j | x =0 (1 ≤ i ≤ r + 1) , (10.18) where ( A in ) 1 ≤ i ≤ r +1 , 0 ≤ n ≤ r is any in vertible matrix. In the W ronskian langua g e, this corresp onds to identifying Q a ( λq a + k ) with X 0 ≤ n 1 < ··· 0 co rresp ond to S U (2 | 2) L and S U (2 | 2) R men tioned earlier . The range m ∈ Z for the “fusion degree” or “string length” for T a,m and Y a,m is a natural conv ention in those systems equipp ed with doubled symmetry , e.g. the O (4) nonlinear sigma mo del ( S U (2) principal chiral ﬁeld) having the global S U (2) L × S U (2) R symmetry [20 2]. 11.3. F orm u l a for planar AdS/CFT sp e ctrum. Now the pla na r AdS/CFT sp ectrum (with R-charge subtra cted) is given in terms of the solutions to the Y- system in the previous subsection by the form ula K 0 X j =1 ǫ 1 ( u 0 ,j ) + X a ≥ 1 Z ∞ −∞ du 2 π i ∂ ǫ ∗ a ( u ) ∂ u ln(1 + Y ∗ a, 0 ( u )) . (11.5) Here K 0 is sp eciﬁed from the sec tor in questio n (see (11.9)–(11.10)) and ǫ a ( u ) is deﬁned b y ǫ a ( u ) = a + 2 ig x ( u + ia 2 ) − 2 ig x ( u − ia 2 ) in terms of x ( u ) s atisfying u g = x ( u ) + x ( u ) − 1 and | x ( u ± ia 2 ) | > 1 . The par a meter g is re la ted to the ’t Ho oft coupling λ b y λ = (4 πg ) 2 . The abov e choice of the branch is calle d ph ysical kinematics. On the other hand, ǫ ∗ a ( u ) w ith a ≥ 1 is deﬁned b y the same fo rmula but with 22 It is essen tiall y the Y-system for U q ( sl (2 | 2)) in Section 2.6. 23 Another, yet more intrinsic wa y of encoding the Y- system together with the T-system is by the quiver in the cluster algebra for mulat i on in Section 5.3. 85 another br anch called mirror kinema tics (cf. [198, 197, 192]). The function Y ∗ a, 0 ( u ) is deﬁned by the mirr or kinematics . Finally , the ra pidities u 0 ,j are determined by the Bethe equation Y 1 , 0 ( u 0 ,j ) = − 1 ( j = 1 , . . . , K 0 ) . (1 1 .6) This description of the pla nar AdS/CFT s p ectr um has been claimed exact for any ’t Ho oft coupling (i.e. to all lo op orders) and op erator s of any ﬁnite L [197, 2 03]. 11.4. Asymptotic Bethe ansatz. T o b e consistent with the ABA equation [1 95], the Y-system (11.2) should split into the left and r ight wings in the limit L → ∞ . Compatibly with this, the middle series sho uld behave as Y a ≥ 1 , 0 ( u ) ≃ x ( u − ia 2 ) x ( u + ia 2 ) ! L φ ( u − ia 2 ) φ ( u + ia 2 ) T L a, − 1 ( u ) T R a, 1 ( u ) , (11.7 ) where φ is a function o b e y ing the relatio n (11.1 5). The las t tw o fa ctors represent the T-functions for the decoupled S U (2 | 2) L and S U (2 | 2) R . They are constructed from the a = 1 case [19, 22] in a w ay ana logous to (9.2), (9 .3) a nd (9.5). Explicitly , the a = 1 case is given as the dressed v acuum for m T L,R 1 , ∓ 1 ( u ) = R (+) 0 ( u − i 2 ) R ( − ) 0 ( u − i 2 ) Q ± 2 ( u − i ) Q ± 3 ( u + i 2 ) Q ± 2 ( u ) Q ± 3 ( u − i 2 ) + Q ± 2 ( u + i ) Q ± 1 ( u − i 2 ) Q ± 2 ( u ) Q ± 1 ( u + i 2 ) − R ( − ) 0 ( u − i 2 ) Q ± 3 ( u + i 2 ) R (+) 0 ( u − i 2 ) Q ± 3 ( u − i 2 ) − B (+) 0 ( u + i 2 ) Q ± 1 ( u − i 2 ) B ( − ) 0 ( u + i 2 ) Q ± 1 ( u + i 2 ) ! , (11.8) where Q l ( u ) = Q K l j =1 ( u − u l,j ). In addition we in tro duce 24 R l ( u ) = K l Y j =1 x ( u ) − x ( u l,j ) p x ( u l,j ) , R ( ± ) l ( u ) = K l Y j =1 x ( u ) − x ( u l,j ∓ i 2 ) q x ( u l,j ∓ i 2 ) , (11.9) B l ( u ) = K l Y j =1 x ( u ) − 1 − x ( u l,j ) p x ( u l,j ) , B ( ± ) l ( u ) = K l Y j =1 x ( u ) − 1 − x ( u l,j ∓ i 2 ) q x ( u l,j ∓ i 2 ) (11.10) for − 3 ≤ l ≤ 3. They are factor ized pieces of Q l ( u ) in that R l ( u ) B l ( u ) = ( − g ) − K l Q l ( u ) , R ( ± ) l ( u ) B ( ± ) l ( u ) = ( − g ) − K l Q l ( u ± i 2 ) . (11.11 ) The num b er s K l sp ecify the relev ant sectors. As usua l in the ana lytic Bethe ansatz (cf. Section 8), analyticity of T L,R 1 , ± 1 ( u ) leads to the equations 1 = Q ± 2 ( u ± 1 ,k + i 2 ) B ( − ) 0 ( u ± 1 ,k ) Q ± 2 ( u ± 1 ,k − i 2 ) B (+) 0 ( u ± 1 ,k ) , 1 = Q ± 2 ( u ± 3 ,k + i 2 ) R ( − ) 0 ( u ± 3 ,k ) Q ± 2 ( u ± 3 ,k − i 2 ) R (+) 0 ( u ± 3 ,k ) , (11.1 2) − 1 = Q ± 1 ( u ± 2 ,k − i 2 ) Q ± 2 ( u ± 2 ,k + i ) Q ± 3 ( u ± 2 ,k − i 2 ) Q ± 1 ( u ± 2 ,k + i 2 ) Q ± 2 ( u ± 2 ,k − i ) Q ± 3 ( u ± 2 ,k + i 2 ) . (11.13) 24 The Bethe r oots u 1 ,j , u 2 ,j , u 3 ,j , u 0 ,j , u − 3 ,j , u − 2 ,j , u − 1 ,j and the T-functions T L 1 , − 1 , T R 1 , 1 here denote u 1 L,j , u 2 L,j , u 3 L,j , u 4 ,j , u 3 R,j , u 2 R,j , u 1 R,j and T L 1 , 1 , T R 1 , 1 in [197], resp ectively . The notation for the Q-functions is also slightly mo diﬁed accordingly . These Bet he roots further correspond to u 1 ,j , u 2 ,j , u 3 ,j , u 4 ,j , u 5 ,j , u 6 ,j , u 7 ,j in [195]. 86 In addition, the cyclicity of the single trace o pe r ator in SYM is to b e reﬂected a s the “zero momentum” co ndition Q K 0 j =1 x ( u 0 ,j + i 2 ) x ( u 0 ,j − i 2 ) = 1. Up on a conv ention adjustment, these relations coincide with the ABA equation in [195, s ection 5.1] except the most complicated one − 1 = x ( u 0 ,k − i 2 ) x ( u 0 ,k + i 2 ) ! L  B (+) 0 B 1 B − 1 R 3 R − 3 /R (+) 0  ( u 0 ,k + i 2 )  B ( − ) 0 B 1 B − 1 R 3 R − 3 /R ( − ) 0  ( u 0 ,k − i 2 ) S ( u 0 ,k ) 2 , (11.1 4) which inv olves the dre s sing factor σ [2 00] via S ( u ) = Q K 0 j =1 σ ( x ( u ) , x 0 ,j ). The ABA equation (11.14) is to b e repro duced in the pre s ent scheme as the lar ge L limit of the equatio n (11.6). In view of T L,R 1 , ∓ 1 ( u 0 ,j ) = − Q ± 3 ( u 0 ,j + i 2 ) Q ± 3 ( u 0 ,j − i 2 ) and (11.7), this amounts to pos tulating that φ therein should satisfy the diﬀer ence equation φ ( u − i 2 ) φ ( u + i 2 ) = B (+) 0 B 1 B − 1 R (+) 0 B 3 B − 3 ( u + i 2 ) R ( − ) 0 B 3 B − 3 B ( − ) 0 B 1 B − 1 ( u − i 2 ) S ( u ) 2 . (11.1 5) The as ymptotics (11.7) with (11.15) sp eciﬁes the la r ge L solution of the Y-system. With r egard to the ﬁnite L eﬀects, the ab ov e formulation repro duces wr apping correctio ns at weak coupling for t wist tw o op erator s obta ined by other methods such as the L ¨ usc her formula. F or insta nce in the ca se of the Ko nishi o p e rator T r( D 2 Z 2 − D Z D Z ), one gets the scaling dimension from ABA as E ABA = 4 + 12 g 2 − 48 g 4 + 336 g 6 − (2820 + 288 ζ (3)) g 8 . The a bove Y- s ystem approach yields the result E ABA + E wrapping with the co rrection E wrapping = (324 + 86 4 ζ (3) − 1 440 ζ (5)) g 8 starting at four-lo op in agre e ment with [19 2]. 11.5. Area of m inimal surface in AdS. Now we turn to the second topic of this section. The T and Y-systems play an essen tia l role in ca lculating the action of classical op en str ing solutions, i.e. the area of minimal surface, in AdS spa ce. Via the AdS/ C FT corr e s po ndence, this yields the planar amplitudes of gluon sc a ttering in N = 4 SYM a t s trong coupling. The gluo n moment a are incorp ora ted in null po lygonal conﬁgurations at the AdS b oundary . The ﬁrst importa nt step in this problem is to linear ize the eq uation of motion of the AdS sigma model (Section 11.5). Once this is achiev ed, the T and Y-sys tems come into the ga me natura lly through the Stok es phenomena of the auxiliar y linear problem ar o und the irr e gular singularity at the b oundary of the worldsheet (Section 11 .6). This part is clo se in spirit to Section 10.1. Extra complication can o ccur when passing to the TBA- t yp e nonlinea r integral equa tions most typically due to the complex nature of the driving terms (“co mplex ma ss” app earing in a symptotics of Y-functions). They ar e determined b y p erio d integrals of the Riemann s urface r eﬂecting the null p olygo nal bo undary and the cross ratios of gluon momenta. The reg ula rized area is fo r mally expressed in the same form as the free energ y in the con ven tional TBA analysis (Section 1 1.8). Sections 11.5 – 11.8 are quic k digest o f these recen t pro gress [204, 205, 206, 207] along a simple version of AdS 3 . The AdS 3 is g iven in terms of the global co or dinate ~ Y = ( Y − 1 , Y 0 , Y 1 , Y 2 ) ∈ R 2 , 2 as ~ Y · ~ Y := − Y 2 − 1 − Y 2 0 + Y 2 1 + Y 2 2 = − 1 . (11.1 6) 87 General pro duct ~ A · ~ B in R 2 , 2 is deﬁned similar ly with the signa ture − 1 , − 1 , 1 , 1. The equation of motion and the Viraso ro co nstraint read ∂ ¯ ∂ ~ Y − ( ∂ ~ Y · ¯ ∂ ~ Y ) ~ Y = 0 , ∂ ~ Y · ∂ ~ Y = ¯ ∂ ~ Y · ¯ ∂ ~ Y = 0 , (11.17) where ∂ = ∂ ∂ z , ¯ ∂ = ∂ ∂ ¯ z and z is a complex co o rdinate par a meterizing the worldsheet. This c la ssical motion of s tr ings in AdS 3 is integrable. In fact, it is transformed to a Z 2 -pro jected S U (2) Hitchin system throug h a Pohlmeyer type reduction [208, 209]. T o see this, introduce the new v ar iables α and p b y e 2 α ( z , ¯ z ) = 1 2 ∂ ~ Y · ¯ ∂ ~ Y , N a = 1 2 ǫ abcd Y b ∂ Y c ¯ ∂ Y d , (11.1 8) p = 1 2 ~ N · ∂ 2 ~ Y , ¯ p = − 1 2 ~ N · ¯ ∂ 2 ~ Y . (11.19) Note that ~ N · ~ Y = ~ N · ∂ ~ Y = ~ N · ¯ ∂ ~ Y = 0 and ~ N · ~ N = 1 . The v a r iable α = α ( z , ¯ z ) is rea l and ~ N is pure ima ginary . Moreover it can b e shown from (1 1.16)-(11.19) that p = p ( z ) is holo morphic. The area is given by 4 R d 2 z e 2 α . The α s a tisﬁes the sinh-Gordon equation modiﬁed with p a s ∂ ¯ ∂ α − e 2 α + | p ( z ) | 2 e − 2 α = 0. As this fact indicates , the equa tions (11.1 7) a re expres sible a s the ﬂatness condition of the connections: ∂ B L ¯ z − ¯ ∂ B L z + [ B L z , B L ¯ z ] = 0 , ∂ B R ¯ z − ¯ ∂ B R z + [ B R z , B R ¯ z ] = 0 , (11.20) where the connections are given by B L z = B z (1) , B L ¯ z = B ¯ z (1) , B R z = U B z ( i ) U − 1 , B R ¯ z = U B ¯ z ( i ) U − 1 , (11 .21) B z ( ζ ) =  1 2 ∂ α − ζ − 1 e α − ζ − 1 e − α p ( z ) − 1 2 ∂ α  , B ¯ z ( ζ ) =  − 1 2 ¯ ∂ α − ζ e − α ¯ p ( ¯ z ) − ζ e α 1 2 ¯ ∂ α  , (11.22) with U =  0 e π i/ 4 e 3 π i/ 4 0  . Here ζ is the sp ectral parameter. Actually the re- lation ∂ B ¯ z ( ζ ) − ¯ ∂ B z ( ζ ) + [ B z ( ζ ) , B ¯ z ( ζ )] = 0 including ζ is s atisﬁed. Splitting the connection in to ζ de p endent part and the rest as B z ( ζ ) = A z + ζ − 1 Φ z and B ¯ z ( ζ ) = A ¯ z + ζ Φ ¯ z , one ﬁnds that the ﬂatness conditions form the Hitchin sys tem with g auge ﬁeld A and Higgs ﬁeld Φ. The g auge group is S U (2) but the system is Z 2 -pro jected in the sense that the a bove form (11.22) b elong s to the inv ariant subspace under the in volution A z → σ 3 A z σ 3 , Φ z → − σ 3 Φ z σ 3 and similarly for A ¯ z and Φ ¯ z . ( σ 3 is a Pauli matrix.) With each zero cur v ature condition in (11.2 0), there is asso ciated a pair of auxil- iary linear pro blems whose co mpatibility yields it. Tha nks to the relations (11.2 1), one can combine and pro mote them into the ζ -dependent versions ( ∂ + B z ( ζ )) ψ = 0 and ( ¯ ∂ + B ¯ z ( ζ )) ψ = 0 or equiv alently ,  d + Φ z dz ζ + A + ζ Φ ¯ z d ¯ z  ψ = 0 (11.23) with A = A z dz + A ¯ z d ¯ z for ψ = ψ ( z , ¯ z ; ζ ). A useful pr op erty is that if ψ ( ζ ) is a ﬂat section with sp ectral parameter ζ , then so is σ 3 ψ ( e π i ζ ) by the Z 2 -symmetry . Given tw o solutions ψ , ψ ′ to (11 .23), deﬁne their S L (2 )- in v ariant pairing a s h ψ , ψ ′ i = ǫ αβ ψ α ψ ′ β , where ψ = ( ψ 1 , ψ 2 ) T , etc. This is a cons tant function o n the worldsheet playing the role analog ous to W rons k ians in Section 1 0. Let ψ L a = ( ψ L 1 ,a , ψ L 2 ,a ) T ( a = 1 , 2) b e the tw o solutions ψ ( z , ¯ z , ζ = 1) normalized as h ψ L a , ψ L b i = 88 ǫ ab . Fix a lso the s o lutions ψ R ˙ a = ( ψ R 1 , ˙ a , ψ R 2 , ˙ a ) T ( ˙ a = 1 , 2) which are similarly nor mal- ized at ζ = i . Then the original AdS 3 co ordinate ~ Y = ( Y − 1 , Y 0 , Y 1 , Y 2 ) is repro duced from the auxiliary linear problem by  Y − 1 + Y 2 Y 1 − Y 0 Y 1 + Y 0 Y − 1 − Y 2  a, ˙ a = ψ L 1 ,a ψ R 1 , ˙ a + ψ L 2 ,a ψ R 2 , ˙ a . (11.24) This substantially achieves the linearizatio n of the pr oblem. 11.6. Stok e s phenomena, T and Y-system . Scattering amplitudes for 2 n glu- ons cor resp ond to o pe n string so lutio ns having po lygonal shap es with 2 n c us ps at the AdS 3 bo undary . This translates to the following b oundar y condition: α → 1 4 ln | p ( z ) | 2 ( z → ∞ ) , p ( z ) = z n − 2 + · · · (po lynomial of degree n − 2) . (11.25) W e assume that n is odd for simplicity . F rom (11.22), s olutions of the auxiliary linear problem (11.23) as | z | → ∞ b ehav e as ψ ∼  ( ¯ p/p ) 1 8 ± ( p/ ¯ p ) 1 8  exp  ± 1 ζ Z √ pdz ± ζ Z √ ¯ pd ¯ z  . (1 1.26) Since exp( 1 ζ R √ pdz ) ∼ exp( z n/ 2 ζ ) holds asymptotically , there are n Stokes sec to rs which are separated by n r ays in the z pla ne. W e lab el them conse c utiv ely a nti- clo ckwise. Let s k ( ζ ) b e the small (sub dominant in the ter minology of Section 1 0) solution in the k th Stokes sector . Then we hav e the pr op erties like σ 3 s k ( e π i ζ ) ∝ s k +1 ( ζ ), s k ( e 2 π i ζ ) ∝ s k +2 ( ζ ) a nd h s j , s k i ( e π i ζ ) = h s j +1 , s k +1 i ( ζ ). Fixing the sma ll solution s 1 ( ζ ) in the ﬁrs t Stokes sector, we deﬁne the others by s k +1 ( ζ ) = ( σ 3 ) k s 1 ( e kπ i ζ ). Set T k ( ζ ) = h s 0 , s k +1 i ( e − π i ( k +1) / 2 ζ ) in the no rmalization h s i , s i +1 i ( ζ ) = 1. Then from the simplest P l ¨ uck er relatio n o r Schouten identit y h s i , s j ih s k , s l i−h s i , s k ih s j , s l i + h s i , s l ih s j , s k i = 0 , one ﬁnds T k ( e πi 2 ζ ) T k ( e − πi 2 ζ ) = T k − 1 ( ζ ) T k +1 ( ζ ) + 1 . (11.2 7) This is a v er sion of the lev el n − 2 restricted T- s ystem for A 1 where the conditio ns T 0 ( ζ ) = 1 and T n − 1 ( ζ ) = 0 are imp os e d 25 . Setting further Y k ( ζ ) = T k − 1 ( ζ ) T k +1 ( ζ ) as usual, one gets the level n − 2 r estricted Y-system (for Y − 1 -v ariables in (2.11)) Y k ( e πi 2 ζ ) Y k ( e − πi 2 ζ ) = (1 + Y k − 1 ( ζ ))(1 + Y k +1 ( ζ )) (11.28) with the b oundary condition Y 0 ( ζ ) = Y n − 2 ( ζ ) = 0 in the k direction. 11.7. Asymptotics, W K B and TBA. As is well known, the relation (11.28) determines the Y-functions eﬀectively only with the information on their analyticity . By the deﬁnition, Y k ( ζ )’s are ana ly tic aw ay fro m ζ ± 1 = 0 wher e they p os sess essential singularities. One ca n deduce the asymptotic b ehavior around them using the WKB approximation regar ding ζ ± 1 as the Planck constant. F or example when ζ → 0 , the solutions of (11.2 3), after a simple similarity tra ns formation making Φ z int o √ p diag (1 , − 1), behav e as exp( ± 1 ζ R √ p dz ) times constant vectors. Thus they ar e well a pproximated by pe r forming the integral along the Sto kes (steep est descent) lines deﬁned by ℑ m ( p p ( z ) dz / ζ ) = 0. At a g eneric p oint in the z plane, there is o ne Stokes line pass ing thro ugh it. Exceptio ns ar e zer os of p ( z ) (turning 25 The latter is a slightly weak er condition than T n − 2 ( ζ ) = 1 in the deﬁnition of Section 2.2. 89 po int s). F rom a sing le zero, there emanate three Stokes lines. They g o tow ar d inﬁnit y a long certain dir ections corresp onding to Stokes sector s or ﬂow into another turning po int . The family of these inﬁnitely ma ny non-cros s ing lines constitute the WKB foliations. See Figur e 3. 3 1 2 1 -3 -2 -1 0 3 2 1 -3 -2 -1 0 Figure 3. Example o f Sto kes lines for p ( z ) = z ( z 2 − 1)( z 2 − 4 ). The left and righ t ﬁgures co rresp ond to arg( ζ ) = 0 and π 3 . 1 , re- sp ectively . Blue lines ar e those emanating from tur ning points. The num b er k sp eciﬁes the Stokes sector where s k is small. F or example, h s 1 , s 2 i ∼ ex p( − 1 ζ R C 1 √ pdz ). The int egral R √ p dz along the red lines a nticlockwise yields asymptotics of ln Y 2 ( ζ ) as ζ → 0 . First cons ide r the ca s e in which the zeros of p ( z ) are a lig ned on the r eal axis. Then o ne obtains the estimate like h s 1 , s 2 i ∼ ex p( − R C 1 √ p dz /ζ ). Ther efore the Y-v ariables (without the normaliza tio n cons traint on s i ) Y 2 k ( ζ ) = h s − k , s k ih s − k − 1 , s k +1 i h s − k − 1 , s − k ih s k , s k +1 i ( ζ ) , Y 2 k +1 ( ζ ) = h s − k − 1 , s k ih s − k − 2 , s k +1 i h s − k − 2 , s − k − 1 ih s k , s k +1 i ( e πi 2 ζ ) (11.29) hav e the asymptotics ln Y 2 k ( ζ ) ∼ Z 2 k ζ + · · · , ln Y 2 k +1 ( ζ ) ∼ Z 2 k +1 iζ + · · · ( ζ → 0 ) , (11.30) where Z k = − H γ k √ p dz is the p erio d in tegral along the cy cle γ k going aro und the k th and ( k + 1)st lar g est zeros o f p ( z ) (cf. Fig. 5 in [206]). The as y mptotics as ζ → ∞ is s imilarly inv es tigated. T ogether with the ζ → 0 case, the result is summa- rized as ln Y k ( e θ ) = − m k cosh θ + · · · ( θ → ±∞ ), where m 2 k = − 2 Z 2 k and m 2 k +1 = 2 iZ 2 k +1 are b oth p ositive. Now that the combination ln( Y k ( e θ ) /e − m k cosh θ ) is an- alytic in the strip |ℑ m θ | ≤ π 2 and decays a s | θ | → ∞ within it, the standard argument leads to the integral equation: ln Y k ( e θ ) = − m k cosh θ + Z ∞ −∞ ln[(1 + Y k − 1 ( e θ ′ ))(1 + Y k +1 ( e θ ′ ))] dθ ′ 2 π cos h( θ − θ ′ ) (11.31) for 1 ≤ k ≤ n − 3 ( Y 0 ( ζ ) = Y n − 2 ( ζ ) = 0). Up to the driv ing (mas s) term, this has the sa me form with the integral equation in TB A or QTM a nalyses asso ciated with the level n − 2 restr ic ted Y-system for A 1 . See for exa mple (15.14) and (16.28). So far , we have co nsidered the case where the zer os of p ( z ) ar e on the rea l axis. When they deviate fro m it, the T and Y-s ystem r emain unchanged. On the 90 other hand, the asymptotics is modiﬁed a s ln Y k ( ζ ) ∼ − m k 2 ζ ( ζ → 0) and ln Y k ( ζ ) ∼ − ¯ m k 2 ζ ( ζ → ∞ ), where m k = | m k | e iϕ k is complex in general. Consequently , the int egral equation (11.31) is replaced with ln ˜ Y k ( e θ ) = − | m k | cos h θ + X j = k ± 1 Z ∞ −∞ ln(1 + ˜ Y j ( e θ ′ )) dθ ′ 2 π cosh( θ − θ ′ + iϕ k − iϕ j ) , (11.3 2) where ˜ Y k ( e θ ) = Y k ( e θ + iϕ k ). This holds for | ϕ k − ϕ k ± 1 | < π 2 . If the phase s go beyond this r ange (so-c a lled wall cr ossing), the integral equatio n acq uir es extra terms corresp o nding to the contributions of the p oles from the conv o lutio n kernel. A simple illustration of such a situation has be en given in [206, app endix B]. 11.8. Area and free ene rgy. The interesting pa rt A of the area is given by 26 A = 2 Z d 2 z T r(Φ z Φ ¯ z ) = i Z √ p dz ∧ Φ 11 ¯ z d ¯ z = − i n − 3 X j,k =1 w j k I γ j √ p dz I γ k Φ 11 ¯ z d ¯ z , (11.33) where the ga ug e Φ z = √ p diag (1 , − 1) is taken and T r Φ ¯ z = 0 is use d. In the last eq uality we hav e dropp ed the co ntribution fro m inﬁnit y . The matrix ( w j k ) is the inv erse of the intersection forms 27 ( h γ j , γ k i ) s pec iﬁe d by h γ 2 k , γ 2 k ± 1 i = 1 . Set ˆ Y 2 k ( ζ ) = Y 2 k ( ζ ) and ˆ Y 2 k +1 ( ζ ) = Y 2 k +1 ( e − πi 2 ζ ) somehow reconciling the shift in (11.29). The factor H γ k Φ 11 ¯ z d ¯ z in (11.33) also app ears as the co eﬃcient of − ζ in the small ζ e x pansion of ln ˆ Y k ( ζ ) based on the pertur bative solution of (11.23). On the other hand, the small ζ = e θ expansion of (11.3 2) g ives ln ˆ Y k ( ζ ) = Z k ζ + ζ h ¯ Z k + X j h γ k , γ j i π i Z dζ ′ ζ ′ 2 ln(1 + ˆ Y j ( ζ ′ )) i + · · · , (11.34 ) where the a pp ea rance of h γ k , γ j i is the eﬀect of using ˆ Y k ( ζ ) ra ther than Y k ( ζ ). Thus one can subs titute H γ k Φ 11 ¯ z d ¯ z in (11 .33) by [ . . . ] her e times ( − 1 ). As the result the area is expressed as A = A p erio ds + A ′ free with A p erio ds = − i X j,k w j k Z k ¯ Z j , A ′ free = − 1 π X k Z k Z dζ ζ 2 ln(1 + ˆ Y k ( ζ )) . (11.35) Actually o ne s hould replace A ′ free by the av era ge A free taking the c o ntribution from large ζ into account. Thus the ﬁnal result reads A = A p erio ds + A free with A free = X k | m k | Z ∞ −∞ dθ 2 π cosh θ ln(1 + ˜ Y k ( e θ )) (11.36) in terms of ˜ Y k ( e θ ) deﬁned after (11.32). This has the s ame for m as the free energy in the conv entional TBA. See for ex a mple (15.15). T o s ummarize, the symmetry as pe c ts of the problem (AdS, Virasor o co nstraints, nu ll-cusp b o unda ry) are incor po rated int o the restric ted T and Y-systems. Then, all the dy namical information (gluo n momen ta, Riema nn surface, cycles) are re- mark ably in teg rated in the “complex mass” para meters m 1 , . . . , m n − 3 . 26 Our Φ z here is ˜ Φ z in [206]. 27 The inv erse exists under our assumption of n b eing o dd. The intersect i on form h , i here should not be confused with the S L (2)-in v ariant pairing of spinors. 91 11.9. Bibli ographical notes. The sub jects in this s ection ar e curr ently in the course o f rapid development. F o r v arious asp ects o f the planar AdS/CFT sp ectrum, see the literatur es given in the end o f Se c tio n 11.1 and r eference therein. W e hav e only dealt with the limited issues r elated to T and Y-systems. The conten ts in Section 1 1.2 – 11.4 are ma inly ba s ed on [197]. F or n umer ical studies, it is imp ortant to formulate the analyticit y precisely and to derive the TBA (or o ther type of ) int egral equations including excited states. W e r efer to [196, 197, 1 9 8, 203, 210] for this pr oblem. Similar analyses ha ve b een made in [21 1, 21 2, 213] for the AdS 4 /CFT 3 duality prop os ed re c e n tly [214]. Calculation of g luon scattering amplitudes a t str o ng coupling using ga uge/string duality was initiated in [20 4] and developed in a series o f w or ks [20 5, 215, 206, 207, 216, 2 17]. F or classica l integrabilit y of AdS sigma mo dels and their connection to Hitc hin system, see also [218]. Auxiliar y linear problem in Section 11.7 is a spec ial case of tha t for g eneral S U (2) Hitchin system [219], where a num b e r o f asp ects in the Riemann-Hilb ert pr oblem have b een dis cussed including WKB triang ulations, the F o ck-Goncharo v co o rdinates, the K ontsevic h-So ib elma n w a ll- crossing formula, TBA and so forth. The co nten ts of Section 11.5 – 11.8 are mainly taken from [206]. W e hav e treated n (num b er of g luons) o dd case. F or the case n even, se e [216, 217]. In [217], further eﬀect of op erato r ins e r tion is studied, and the (slig h tly deformed) level 2 restricted Y-system for D n has b een obtained. F or a similar app ear a nce of the D type Y-sy s tem in A 1 related lattice mo dels , see Remark 16.8. The genera lized sinh-Gordon equatio n has also b een studied in the context of g eneralized ODE/ I M corres p o ndence in [175]. 12. Aspects as classical integrable system Besides the quantum integrable systems, T and Y-systems also ha ve in terest- ing a sp ects a s classical nonlinear diﬀerence equations. F o r instance, the T-sys tem relation (2.5) is presented in the for m τ 1 τ 23 − τ 2 τ 31 + τ 3 τ 12 = 0 (12.1) with a suitable redeﬁnition up to the b oundary conditio n. Here the indices sig nify a shift of the indep endent vector v aria ble in the respective directions ( τ ij = τ j i ). This is a version of Hirota-Miwa equation on ta u functions in the theor y of discrete KP equations [2 20, 221, 22 2, 223]. A simplest account for its in tegrability is the Lax representation, namely , the compatibility of the linear system: ψ i − ψ j = τ τ ij τ i τ j ψ ( i < j ) . The Hirota-Miwa equa tion ser ves as a mas ter equa tion genera ting a v a riety of soliton equations under suitable s p ecia lizations and b oundar y co nditions. See for instance [22 2, 22 4, 22 5]. Apart from this, there are n umero us asp ects in type A T-system, sometimes called o ctahedron recurrence , related to discrete geometry [226, 227, 22 8], Littlewo o d-Richardson rule [229], p erfect matchings and par tition functions on a netw ork [230, 231] and so forth. F o r types other than A how ever, such results are re latively few. Our presentation in this section is necessar ily selective. In Section 12.1, we ex- plain that the T-system for g is a discr e tized T o da ﬁeld equation that has decent contin uous limits with a known Hamiltonian structur e. In Section 12.2, a connec - tion of the Y-system for A ∞ with discre te geo metry is r eviewed. 92 12.1. Cont in uum limit. W e present a simple contin uous limit o f the T-sys tem for general g known a s the lattice T oda ﬁeld equation [232]. It is a diﬀerence-diﬀerential system co nt aining contin uous time a nd discrete space v ariables. F urther c ontin uous limit on the latter yields the T o da ﬁeld e q uation on (1 + 1)-dimens ional co n tin uous spacetime [233]. W e b egin by ma king a slight change of v ar iables in the T-sys tem as T ( a ) m ( u ) = τ a ( u + m t a , s + ε m t a ) (1 ≤ a ≤ r, u ∈ Z /t, m ∈ Z ) . (12.2) Here ε is a sma ll parameter and s is going to b e the contin uous time v aria ble so on. F or the symbols t, t a and ro ot s ystem data, se e around (2 .1). W e substitute (12.2) into the T-system (2.22) T ( a ) m ( u − 1 t a ) T ( a ) m ( u + 1 t a ) − T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) = g ( a ) m ( u ) M ( a ) m ( u ) with m ∈ t a Z . F or each g of rank r there are r such equatio ns. (The case m 6∈ t a Z leads to the same contin uum limit as the one considered in the following.) F or example, the B 2 case reads τ 1 ( n − 1 , s ) τ 1 ( n + 1 , s ) − τ 1 ( n − 1 , s − ε ) τ 1 ( n + 1 , s + ε ) = g 1 τ 2 ( n, s ) , τ 1 ( n − 1 2 , s ) τ 1 ( n + 1 2 , s ) − τ 1 ( n − 1 2 , s − ε 2 ) τ 1 ( n + 1 2 , s + ε 2 ) = g 2 τ 1 ( n − 1 2 , s ) τ 1 ( n + 1 2 , s ) , where we hav e chosen g a = g ( a ) t a m ( u ) to b e a c o nstant. W e take the cont inu um limit in the time v ariable s keeping n ∈ Z /t a s the co ordinate of a o ne dimensional lattice without boundar y . Namely , w e replace g a by εg a /t a and set ε → 0. The result reads D s τ 1 ( n − 1) · τ 1 ( n + 1) = g 1 τ 2 ( n ) , D s τ 2 ( n − 1 2 ) · τ 2 ( n − 1 2 ) = g 2 τ 1 ( n − 1 2 ) τ 1 ( n + 1 2 ) . Here we suppressed the time dep endence as τ a ( n ) = τ a ( n, s ), whic h we shall als o do in the remainder of this subsection. D s denotes the Hirota deriv ative: D s f · g = ∂ f ∂ s g − f ∂ g ∂ s . Similarly , the gener al g ca se is given by D s τ a ( n − 1 t a ) · τ a ( n + 1 t a ) = g a M a ( n ) , M a ( n ) := Y b : C ab = − 1 τ b ( n ) Y b : C ab = − 2 τ b ( n − 1 2 ) · τ b ( n + 1 2 ) Y b : C ab = − 3 τ b ( n − 2 3 ) τ b ( n ) τ b ( n + 2 3 ) , (12.3) where n ∈ Z /t . W e ca ll this the lattice T o da ﬁeld equation for g . In some cas e, it ac tually splits in to disjoint sectors. F or instance in types ADE , o ne ha s t a = t = 1 for a ny a ∈ I , hence (12 .3) clo s es among { τ a ( n ) | a ∈ I ( − 1) n } o r { τ a ( n ) | a ∈ I ( − 1) n +1 } , where I ± is the bipartite decomposition of the Dynkin diag ram no des I = { 1 , . . . , r } = I + ⊔ I − . One can rewrite (1 2.3) in a for m that lo ok s more like T o da equatio n a nd explo r e its Ha milto nia n structure. As an illustration, we ﬁr st tr e at the A 1 case. Let us int ro duce the dynamical v ariables x ( n ) and β ( n ) by x ( n ) = ∂ ∂ s ln τ 1 ( n − 1) τ 1 ( n + 1) , β ( n ) = x ( n − 1) x ( n + 1) ( n ∈ Z ) . (12.4) 93 Equation (12.3) for A 1 reads ∂ τ 1 ( n − 1) ∂ s τ 1 ( n + 1) − τ 1 ( n − 1) ∂ τ 1 ( n + 1) ∂ s = g 1 . (12.5) This allows us to rewrite (12.4) as x ( n ) = g 1 τ 1 ( n − 1) τ 1 ( n + 1) , β ( n ) = τ 1 ( n + 2) τ 1 ( n − 2) . (12.6 ) F rom the expres s ion o f x ( n ) in (12.4) and β ( n ) in (12.6), one gets ano ther form of the lattice T o da ﬁeld equation for A 1 : ∂ ln β ( n ) ∂ s = − x ( n − 1) − x ( n + 1) , (12.7) which is a discrete analo g of the Liouville equatio n. It is derived as the eq ua tion of motion ∂ β ( n ) ∂ s = {H , β ( n ) } , (12.8) with the following Hamiltonian a nd P oisson brack et: H = X m ∈ Z x ( m ) , { x ( m ) , x ( n ) } = x ( m ) x ( n ) s gn 2 ( n − m ) . (12.9) See (12 .1 3) for the deﬁnition of sg n 2 ( n ). W e rema rk that (12.5), (1 2.7) a nd their relation explained in the ab ov e are diﬀerence- diﬀerential a nalog of the T- system, Y-system and their transfor ma tion sta ted in Theorem 2.5 for A 1 , resp ectively . All these features are g eneralized to g straightforwardly . The re le v ant dynamical v ariables are x a ( n ) , β a ( n ) = x a ( n − 1 t a ) x a ( n + 1 t a ) ( a ∈ I , n ∈ Z /t ) , (12.1 0) which ar e functions of the contin uous time s . W e keep the nota tion I , t, t a , C , ( α a | α b ) around (2.1) and set B ab = B ba = t b max( t a , t b ) C ab =      2 C ab = 2 , − 1 C ab < 0 , 0 C ab = 0 . (12.11) ( B ab ) is the Cartan matrix fo r s imply laced Dynkin diagra m o btained b y forg etting the m ultiplicit y of oriented edges in tha t fo r g . W e sp ecify the Poisson bracket of x a ( n ) as { x a ( m ) , x b ( n ) } = 1 2 B ab x a ( m ) x b ( n ) sgn B ab  max( t a , t b )( n − m )  , (12 .12) where sgn k ( v ) with k ∈ { 2 , − 1 } is the o dd function of v ∈ R deﬁned by 28 sgn k ( v ) =      1 if v > 0 and v ∈ 2 Z + k , − 1 if v < 0 a nd v ∈ 2 Z + k , 0 other wis e . (12.13) 28 sgn 0 ( v ) is not necessary since the RHS of (12.12) cont ains the factor B ab . 94 Consequently , the Poisson brack et concer ning β a ( n ) b ecomes loca l in tha t it is non v anishing only with ﬁnitely many opp onents. { x a ( m ) , β b ( n ) } =        − x a ( m ) β a ( n )( δ m,n + 1 t a + δ m,n − 1 t a ) C ab = 2 , x a ( m ) β b ( n ) P − C ab − 1 j = C ab +1 δ m + j t a ,n C ab < 0 , 0 C ab = 0 , (12.14) { β a ( m ) , β b ( n ) } = β a ( m ) β b ( n )( δ m +( α a | α b ) ,n − δ m − ( α a | α b ) ,n ) . ( 12.15) In (12.14), the j -sum is taken with the condition j ≡ C ab + 1 mod 2. The eq uation of motion with the Hamiltonian ∂ β a ( n ) ∂ s = {H , β a ( n ) } , H = X a ∈ I ,n ∈ Z /t x a ( n ) ( 12.16) leads to the diﬀerential-diﬀerence s y stem: ∂ ln β a ( n ) ∂ s = − x a ( n − 1 t a ) − x a ( n + 1 t a ) + X b : C ba = − 1 x b ( n ) + X b : C ba = − 2  x b ( n − 1 2 ) + x b ( n + 1 2 )  + X b : C ba = − 3  x b ( n − 2 3 ) + x b ( n ) + x b ( n + 2 3 )  . (12.17) F or g = A 1 this reduces to (12.7). The equation (12.1 7) with x a ( n ) and β a ( n ) related as (12.1 0) is another form o f the la ttice T o da ﬁeld equation (12.3). In fact, the tr a nsformation b etw een (12.3) and (12.1 7) is parallel with the A 1 case (12.4)–(12.7). Generalizing (12 .4) w e r e la te x a ( n ) and τ a ( n ) by x a ( n ) = ∂ ∂ s ln τ a ( n − 1 t a ) τ a ( n + 1 t a ) = g a M a ( n ) τ a ( n − 1 t a ) τ a ( n + 1 t a ) , (12.18) where the latter equality is due to the lattice T o da ﬁeld equation (12.3). Substi- tuting the latter form into (12.1 0), w e ﬁnd β a ( n ) = Y b ∈ I τ b ( n + ( α a | α b )) τ b ( n − ( α a | α b )) . (12.1 9) This can also b een derived from (8.1 6) by no ting the same structure in A − 1 a,z = q tn (4.25) and M a ( n ) / ( τ a ( n − 1 t a ) τ a ( n + 1 t a )) given by (12.3). Anyw ay , ∂ ln β a ( n ) ∂ s is expressed as a linea r co m bination of x a ( n ) by using the ﬁrst formula in (12.18). The result repro duces (12.17). A further contin uous limit on n can b e taken b y letting x a ( n ) → 2 ε exp( φ a ( z + εn )) , ln β a ( n ) → − 2 ε t a φ ′ a , (12 .20) where ′ = ∂ ∂ z . Then the limit ε → 0 of (1 2.17) leads to a version of the T o da ﬁe ld equation for φ a = φ a ( z , s ): ∂ 2 φ a ∂ z ∂ s = X b ∈ I t a t b ( α a | α b ) e φ b . (12.21) 95 The case g = A 1 is the Lio uville equation. Switching to ψ a by φ a = P b ∈ I C ab ψ b − ln t a , one may rewr ite it in the form ∂ 2 ψ a ∂ z ∂ s = exp  X b ∈ I C ab ψ b  studied in [233]. An explicit constr uction of the genera l solutio n is known containing 2 r arbitrar y functions [233]. W e see that (12.16) and (12 .14) ar e lattice a nalog of the Hamiltonian formulation o f the T o da ﬁeld equation: ∂ φ ′ a ∂ s = {H , φ ′ a } , H = X a ∈ I Z dz e φ a ( z ) , { φ a ( z ) , φ ′ b ( z ′ ) } = t a t b ( α a | α b ) δ ( z − z ′ ) . The Poisson structures (12.1 2)–(12.15) ha ve an origin in the lattice analog of the W -algebra s go ing back to [23 4]. In par ticular, they may be deduced from the Poisson relations among appro priate constituent ﬁelds in the q -defo rmed W - algebra. See for example [2 3 5, 2 36, 154, 232, 237] and reference ther e in. Here we only mention, as an example, that (1 2.15) is a la ttice ana log o f the Poisson relation { A a ( z ) , A b ( w ) } =  δ  q ( α a | α b ) w z  − δ  q − ( α a | α b ) z w  A a ( z ) A b ( w ) among the ﬁelds A a ( z ) corresp o nding to the exp onential simple ro o t e α a whose counterpart in the theo ry of q -character ha s app eared in (4.25). See equation (3.1 ) in [237] and also equation (8.8) in [70] for the log arithmic form. 12.2. Discrete geometry. As we hav e see n in the prev ious subsection, co n tin uous limits of T- system lead to T o da type diﬀerential eq ua tions. O n the o ther hand, geometric o rigins of ma ny diﬀerential equatio ns of s uch kind have b een known from the days of Darb oux. Like the contin uo us cas e, it is na tural to seek discrete geometry resp o ns ible for the integrability of disc rete integrable equations. In fact, if w e let such g eometric ob jects sp eak of themselves, they w ould say “W e exist, therefore it is integrable 29 ”. There a re many results in this direc tio n. See for example [23 8, 22 6, 22 7, 22 8, 23 9] and reference ther ein. In a sense they pro vide a most na tural fra mework to set up Lax for malisms of the integrable diﬀerence equations from g e o metric points of view. Here we only include a simple exp ositio n of the ba sic example [240, 241] co nnecting Y-system for A ∞ to a discrete analo g of the Laplace sequence of conjugate nets. W e begin b y recalling the appear ance of the T o da ﬁeld equation in pro jective diﬀerential g eometry . Consider a sur face in the re a l pro jective spa ce P 3 which has the homog eneous co ordinate vector z = z ( x, y ) ∈ P 3 . A lo cal co ordinate ( x, y ) of the surface is called a c onjugate net if z xy + a ( x, y ) z x + b ( x, y ) z y + c ( x, y ) z = 0 (12 .22) is v a lid for so me functions a, b, c , where the indices mea n the der iv atives. Although z and w specify the same surface if they are related by z = λ w , the ab ove equa tion is not inv aria nt but changed in to w xy + ˜ a ( x, y ) w x + ˜ b ( x, y ) w y + ˜ c ( x, y ) w = 0 (12.23) 29 V. V. Bazhano v, talk at Newton Institute, Cambridge, UK, March 2009. 96 with ˜ a = a + (ln λ ) y , ˜ b = b + (ln λ ) x , ˜ c = c + a (ln λ ) x + b (ln λ ) y + λ xy /λ . A characteristic of a surfa c e indep endent of the gauge λ is the Laplace inv ar ia nt h = a x + ab − c, k = b y + ab − c, (12 .24) satisfying ˜ h = h and ˜ k = k . In what follows w e consider the gener ic situation that they are nonzero. F or the ho mo geneous co or dina te vector z sa tisfying (1 2.22), the La pla ce trans- formation L ± is deﬁned by L + ( z ) = z y + a z , L − ( z ) = z x + b z . (12.25) This is compatible with the deﬁning pro p erty (12.22) of the conjuga te net in that L + ( λ w ) = λ ( w y + ˜ a w ) and L − ( λ w ) = λ ( w x + ˜ b w ) hold with ˜ a and ˜ b g iven in the ab ov e equation. Any comp onent z of z trans forms a s L − ◦ L + ( z ) = hz and L + ◦ L − ( z ) = k z , meaning that L + and L − are inv erse to ea ch o ther a s transformatio ns in P 3 . The family of surfaces in P 3 generated from z (0) = z as z ( ± n ) = ( L ± ) n ( z ) ( n ≥ 1) is called a Laplace sequence. Denote by h n , k n the Laplace inv ar iant asso ciated with z ( n ) . It is easy to see that z ( ± 1) satisﬁes (12 .22) with a, b, c replace d by a ( ± 1) , b ( ± 1) , c ( ± 1) given by a (1) = a − h y h , b (1) = b, c (1) = ab − h + h  b h  y , a ( − 1) = a, b ( − 1) = b − k x k , c ( − 1) = ab − k + k  a k  x . (12.26) Substituting this in to (12.24), one can express h ± 1 and k ± 1 in terms of h 0 = h a nd k 0 = k . The result shows that the s e quence of Lapla ce inv aria nt s sa tisfy a T o da ﬁeld equation for A ∞ : ∂ 2 ln h n ∂ x∂ y = − h n − 1 + 2 h n − h n +1 , h n = k n +1 . ( 12.27) Now w e move onto the discrete analog of these c o nstructions. The ﬁrst step is to observe that (12.22) implies the four inﬁnitesimally neig hboring p oints are coplanar. This motiv ates us to introduce a map x : Z 2 → P 3 such that the 4 p oints x ( n, m ) , x ( n + 1 , m ) , x ( n, m + 1) , x ( n + 1 , m + 1) ar e coplanar for a ny ( n, m ) ∈ Z 2 . Such a map is calle d tw o dimensiona l quadrilater al lattic e , which serves a s a dis crete analog of the conjugate net. I n the inhomogeneous co ordinate of the pro jectiv e space, a tw o dimensional qua drilateral lattice is represented by a map x : Z 2 → R 3 satisfying the discrete analo g o f (12.22) as follo ws: ∆ 1 ∆ 2 x = ( T 1 A )∆ 1 x + ( T 2 B )∆ 2 x. (12.28) Here ∆ i = T i − 1 and T i changes n i in a ny function f ( n 1 , n 2 ) to n i + 1. The functions A, B on Z 2 are “gauge p otentials” analogo us to a, b in the contin uum case. The Laplace trans fo rmation, denoted b y the s ame symbol as be fore, reads L + ( x ) = x − ∆ 1 x B , L − ( x ) = x − ∆ 2 x A . (12.29) T o see the geometric meaning of this, note tha t the fo ur p oints x, T 1 x, T 2 x, T 1 T 2 x form a quadrila ter al on a plane due to (1 2.28). The p oints T 1 L + ( x ) and T 2 L − ( x ) are in ter sections of the t wo lines ex tending the o ppo site sides of the quadrilater al. 97 x T 1 x T 2 x T 1 T 2 x T 1 L − ( x ) T 2 L + ( x ) As in (12.26), the po stulate ∆ 1 ∆ 2 L ± ( z ) = T 1 L ± ( A )∆ 1 L ± ( z )+ T 2 L ± ( B )∆ 2 L ± ( z ) ﬁxes the Laplace transformation of the gaug e p otentials as L + ( A ) = B T 2 B (1 + T 1 A ) − 1 , L + ( B ) = T − 1 2  T 1 L + ( A ) L + ( A ) (1 + B )  − 1 , L − ( A ) = T − 1 1  T 2 L − ( B ) L − ( B ) (1 + A )  − 1 , L − ( B ) = A T 1 A (1 + T 2 B ) − 1 . (12.30) It follows that the Laplac e transforma tion is inv er tible, i.e . L + ◦ L − = L − ◦ L + = id. Int ro duce the Laplace seq uence a s the con tinuous case b y x (0) = x and x ( ± n ) = ( L ± ) n ( x ) ( n ≥ 1). Now w e a re going to assign a c r oss r atio to each mem b er o f the Laplace sequence. F or the four colinea r p oints q 1 , q 2 , q 3 , q 4 in R 3 , we deﬁne the cr oss ratio as cr( q 1 , q 2 , q 3 , q 4 ) = cr( q 2 , q 1 , q 4 , q 3 ) = ( q 3 − q 1 )( q 4 − q 2 ) ( q 3 − q 2 )( q 4 − q 1 ) , which is inv ariant under pr o jective transfor mations. Deﬁne the sequence of cross ratio b y Y ( n ) = − cr( x ( n ) , L + ( x ( n ) ) , T 1 x ( n ) , T 2 L + ( x ( n ) )) ( n ∈ Z ) , (12.31 ) or equiv alently , b y setting Y (0) = Y and Y ( ± n ) = ( L ± ) n ( Y )( n ≥ 1) with Y (0) = Y = − cr( x, L + ( x ) , T 1 x, T 2 L + ( x )). The four p o in ts in cr ar e colinear. By us ing (12.28)–(12.30) one can derive v ario us formulas, e.g. Y = T 2 B − (1 + T 1 A ) B (1 + B )(1 + T 1 A ) = − L + ( A ) 1 + L + ( A ) B 1 + B , Y ( − 1) = − cr( x, L − ( x ) , T 2 x, T 1 L − ( x )) . The sequence Y ( n ) satisﬁes the functional relation [24 0, 241] ( T 1 T 2 Y ( n ) ) Y ( n ) = T 1  1 + Y ( n − 1) 1 + ( Y ( n ) ) − 1  T 2  1 + Y ( n +1) 1 + ( Y ( n ) ) − 1  . (12.32) With a suitable identiﬁcation, this co incides with the Y-system for A ∞ (2.11) Y ( a ) m ( u − 1) Y ( a ) m ( u + 1) = (1 + Y ( a − 1) m ( u ))(1 + Y ( a +1) m ( u )) (1 + Y ( a ) m − 1 ( u ) − 1 )(1 + Y ( a ) m +1 ( u ) − 1 ) with no bo undary conditions on a and m . 12.3. Bibli ographical no tes. The conten ts of Section 1 2.1 and Sectio n 12.2 a re mainly taken from [232, 23 7] and [2 40, 241], resp ectively . 98 13. Q-system and Fermionic formula 13.1. In tro duction. Consider the T-system for g . If one fo rmally forg ets the sp ectral pa rameter u in T ( a ) m ( u ), the resulting v aria ble is conv entionally denoted by Q ( a ) m and the T-system r educes to the re lation among them called Q-system . In the context of q -characters, T ( a ) m ( u ) is the q -character χ q ( W ( a ) m ( u )) of the Kirillov- Reshetikhin mo dule W ( a ) m ( u ) (Theorem 4.8). Therefore, Q ( a ) m = res T ( a ) m ( u ) (13.1) is the usua l character of g o btained by the restrictio n deﬁned in (4.23). Consider an arbitra r y product of Q ( a ) m ’s and the tw o kinds of decomp o sitions (w e assume ν ( a ) m ∈ Z ≥ 0 for the time b eing) Y a,m ( Q ( a ) m ) ν ( a ) m = X λ b λ χ ( V λ ) = X λ c λ e λ . (13.2) Here χ ( V λ ) deno tes the (usual) character o f the ir reducible g - mo dule V λ with hig hest weigh t λ . The multiplicities b λ of the irr educible repres ent ation V λ (branching co eﬃcients) a nd the multiplicities c λ of weigh ts λ (dimensions of weight spac es) are t wo ba sic q ua nt ities characterizing the decomp ositions . It turns out that analyses of the Q -system pro vide them with F ermionic formulas b λ = M λ and c λ = N λ . They p oss ess fa s cinating forms that symbolize the formal c o mpleteness o f the st ring hyp othesis in the Bethe ansatz at q = 1 and q = 0, resp ectively . In Sectio ns 13.2 and 13.3 w e explain how M λ and N λ emerge f rom the Bethe ansatz alo ng the simplest setting in g = A 1 . Precise statements for A 1 are for m u- lated in Section 13.4 and the pro of by a uniﬁed persp ective of the m ultiv aria ble Lagra ng e inv ersion metho d is outlined in Section 13.5. All the essential ingredients are given b y this point. In Section 13 .6, we intro duce the Q-system for g a nd write down the asso ciated F ermionic for m ulas M λ and N λ . The main Theor em 13.11 in the ge neral case is stated. In Section 13.7, the expa nsion of Q ( a ) m int o classical char- acters is given for non exceptio nal a lgebras A r , B r , C r and D r . There are a lo t of further asp ects which a re b eyond the sco pe of this review. They will b e mentioned brieﬂy in Section 1 3 .8. F or simplicit y we restr ict our selves to unt w is ted aﬃne Lie algebras in this section. Analog o us r esults are also av ailable in the twisted cases. 13.2. Simpl est e xampl e of M λ . Recall the Bethe equation (8.4) for the 6 vertex mo del. In the rationa l limit q → 1, it takes the form −  u j + √ − 1 u j − √ − 1  L = n Y k =1 u j − u k + 2 √ − 1 u j − u k − 2 √ − 1 , (13.3) where we hav e set all the inho mogeneity w j = 0 a nd r eplaced u j by √ − 1 u j . The string hypothes is [1 0 ] is that the r o ots u 1 , . . . , u n are arrange d as (called originally “W ellenKomlex ” in [10]) [ m ≥ 1 [ 1 ≤ α ≤ N m [ u mα ∈ R { u mα + √ − 1( m + 1 − 2 i ) + ǫ mαi | 1 ≤ i ≤ m } (13.4) for each par tition n = P m ≥ 1 mN m ( N m ∈ Z ≥ 0 ). Here ǫ mαi stands for a small deviation. The m -tuple conﬁguration (with negligible ǫ mαi ) is called a length m string with string cen ter u mα . The N m is the num b er of length m string s. The string h yp o thesis is not literally true as exempliﬁed for insta nce when n = 2 and 99 L > 21 (cf. [242]). Nev e r theless, a formal co unt of the n um ber of solutions to (13.3) is done a s follows [1 0, 243]. First o ne rewrites the Bethe equation into the one for the str ing cen ters. This is done by repla cing u j by a member o f a string u mα + √ − 1( m + 1 − 2 i ) + ǫ mαi and taking the pro duct over 1 ≤ i ≤ m . The resulting equation in the log arithmic form ln(LHS / RHS) ∈ 2 π √ − 1 Z is cast, if ǫ mαi is negligible, int o the form f m ( u mα ) ∈ Z or Z + 1 2 (1 ≤ α ≤ N m ) whic h dep ends on m and the partition { N m } . Explicitly , f m ( u ) is given by f m ( u ) = L θ m, 1 ( u ) − X k ≥ 1 N k X β =1 ( θ m,k − 1 + θ m,k +1 )( u − u kβ ) , (13.5 ) θ m,k ( u ) = 1 π min( m,k ) X α =1 tan − 1  u | m − k | + 2 α − 1  . (13.6) Let us employ the principal branch − π 2 ≤ tan − 1 ( u ) ≤ π 2 . Then fro m θ m,k ( ±∞ ) = ± min( m, k ) / 2 a nd ( θ m,k − 1 + θ m,k +1 )( ±∞ ) = ± (min( m, k ) − δ m,k / 2), we get f m ( ±∞ ) = ± ( P m + N m ) / 2. Here P m , called v acancy num b er, is given by P m = L − 2 X k ≥ 1 min( m, k ) N k , (13.7) and will play a signiﬁcant role in the sequel. The bold argument is then that if P m ≥ 0, the solutions { u mα } (up to pe r mut ations of u m 1 , . . . , u mN m for ea ch m ) are in one to one corr esp ondence with the sequence s ( I 1 , . . . , I N m ) ∈ ( Z + P m + N m +1 2 ) N m such that − f m ( ∞ ) + 1 2 ≤ I 1 < · · · < I N m ≤ f m ( ∞ ) − 1 2 . There are  P m + N m N m  such sequences for each m . Accordingly if one admits the argument, the num b er of solutions is M n = X { N m } Y m ≥ 1  P m + N m N m  , (13.8) where the s um extends over a ll the partitions o f n , namely those N m ≥ 0 satisfying n = P m ≥ 1 mN m . (W e understand M 0 = 1.) What num b er should we ex pec t for M n ? T he quan tum space for the rational 6 vertex model is ( V ω 1 ) ⊗ L , where V ω 1 ≃ C 2 is the s pin 1 2 representation whose highest weight is the fundamen tal w eig ht ω 1 . As a result of the global A 1 = sl 2 symmetry , the Bethe vectors b eco me by constr uction highes t weight vectors in the quantum s pa ce [2 44]. The sec tor labeled by n ca r ries the weigh t ( L − 2 n ) ω 1 . Thus for the Bethe’s s tr ing hypothesis to b e complete, one should have M n = b n for 0 ≤ n ≤ L/ 2, where b n is the branching co eﬃcient in the irre ducible deco mpo sition ( V ω 1 ) ⊗ L = L 0 ≤ n ≤ L/ 2 b n V ( L − 2 n ) ω 1 30 . E xplicitly , b n =  L n  −  L n − 1  . Note that the condition 0 ≤ n ≤ L/ 2, and (13 .7) imply that P 1 ≥ P 2 ≥ · · · ≥ P ∞ = L − 2 n ≥ 0, which automatically gua rantees the condition P m ≥ 0. Example 13.1. F or L = 6, one has ( V ω 1 ) ⊗ 6 = V 6 ω 1 ⊕ 5 V 4 ω 1 ⊕ 9 V 2 ω 1 ⊕ 5 V 0 . Accord- ingly one can check ( M 0 , M 1 , M 2 , M 3 ) = (1 , 5 , 9 , 5). In fact, the no n trivial cases a re 30 This argumen t lacks the consideration on the asso ciated Bethe vectors. 100 chec ked as M 1 =  4 + 1 1  N 1 =1 = 5 , M 2 =  2 + 1 1  N 2 =1 +  2 + 2 2  N 1 =2 = 9 , M 3 =  0 + 1 1  N 3 =1 +  2 + 1 1  0 + 1 1  N 1 = N 2 =1 +  0 + 3 3  N 1 =3 = 5 . W e p ostp one what ca n b e pr oved mathematica lly in a more genera l setting to Section 13.4. 13.3. Simpl est example of N λ . Here we retur n to the trigonometr ic Bethe equa - tion (8.4). After setting the inhomog eneit y w j = 0, q = e − 2 π ~ and replacing u j by u j / ( √ − 1 ~ ), it reads sin π  u j + √ − 1 ~  sin π  u j − √ − 1 ~  ! L = − n Y k =1 sin π  u j − u k + 2 √ − 1 ~  sin π  u j − u k − 2 √ − 1 ~  . (13.9 ) In this conv ention, the analog of the str ing conﬁguration (13.4) is [ m ≥ 1 [ 1 ≤ α ≤ N m [ u mα ∈ R { u mα + √ − 1( m + 1 − 2 i ) ~ + ǫ mαi | 1 ≤ i ≤ m } , (13.10) where N m is again the num b er of length m s tr ings. Apart from q = 1 treated in the prev io us subsection, there is a p oint q = 0, i.e. the limit ~ → ∞ wher e one ca n make another forma l but systematic counting of the string solutions [245]. Leaving the precise deﬁnitions and statements to [245], we just sta te here casually that at q = 0 the Bethe equation (1 3.9) b ecomes the following linea r congr uence equation on the string centers: X k ≥ 1 N k X β =1 A mα,kβ u kβ ≡ P m + N m + 1 2 mo d Z . (13 .11) Here the co eﬃcient A mα,kβ is given by A mα,kβ = δ mk δ αβ ( P m + N m ) + 2 min( m, k ) − δ mk (13.12) with the same P m as in (13.7). Equation (13.1 1) is called the string center equation. The concrete form of its RHS will not matter in the counting pro blem considered in what follows. Given a string pattern ( N m ), one should actually r egard the so lutions to (13.11) as belo ng ing to ( u k 1 , u k 2 , . . . , u kN k ) ∈ ( R / Z ) N k / S N k for each k , w he r e S N denotes the degree N symmetric g roup. This is beca use the Bethe vector is a sy mmetric function of e 2 π √ − 1 u k 1 , . . . , e 2 π √ − 1 u kN k for each k . W e s ay that a solution ( u kβ ) to (13.11) is oﬀ-diagonal if u k 1 , u k 2 , . . . , u kN k ∈ R / Z are all distinct for each k . This deﬁnition is motiv ated by the fact that the Bethe vectors v anish unless the asso c ia ted Bethe roo ts are all distinct [246]. F or 0 ≤ n ≤ L/ 2 w e deﬁne N n = X { N m } ♯ { oﬀ-diago na l so lutions to the string center eq.(13 .11) } , (13.1 3) where the sum is taken ov er N m ∈ Z ≥ 0 satisfying n = P m ≥ 1 mN m as in (13.8). (W e understand N 0 = 1.) 101 Example 13.2. W e derive N n =  L n  for n = 1 , 2 as an illustr ation. When n = 1 , the o nly p oss ible string pattern ( N m ) is N m = δ m 1 . The equation (13.11) is just Lu 11 ≡ const mo d Z ; hence, there are N 1 = L oﬀ-diag onal so lutions. F or n = 2 (hence L ≥ 4), there are tw o po ssible s tring pa tterns (i) N m = δ m 2 and (ii) N m = 2 δ m 1 . In (i), equation (13 .11) is Lu 21 ≡ const mod Z , which aga in yields L oﬀ-diagona l s olutions. In (ii), equation (13 .11) rea ds in the matrix notatio n as  L − 1 1 1 L − 1   u 11 u 12  ≡ ~ c mo d Z 2 for some ~ c . The num b er of solutions eq uals the determinant L ( L − 2) of the co ef- ﬁcient matrix, which is p ositive by the assumption L ≥ 4. How e ver, they contain the c ollision ( u 11 = u 12 ) L times which should b e excluded from the oﬀ-diag onal solutions. Thus there are ( L ( L − 2) − L ) / 2 oﬀ-diago nal so lutions fo r (ii), whe r e the division by 2 is due to the identiﬁcation by S 2 . Collecting the co ntributions from (i) and (ii), one gets N 2 = L + ( L ( L − 2) − L ) / 2 = L ( L − 1) / 2 a s desired. It is p oss ible to gener a lize the calculatio ns in Example 13.2 by a systematic application of the inclusion-exclusio n principle . The ﬁnal result re a ds [2 45] N n = X { N m } det m,k ∈J ( F m,k ) Y m ∈J 1 N m  P m + N m − 1 N m − 1  , F m,k = δ mk P m + 2 min( m, k ) N k , (13.14) where J = { j ∈ Z ≥ 1 | N j ≥ 1 } a nd P m is deﬁned by (13 .7). Again the sum in (13.14) is taken in the same way a s (13 .1 3). As noted befor e Exa mple 1 3.1, the assumption 0 ≤ n ≤ L/ 2 implies P m ≥ 0 ( m ≥ 1). B y using this pr op erty it can be shown that det m,k ∈J ( F m,k ) > 0 a nd the RHS of the ﬁrs t equality in (13.14) is a po sitive integer. What num b er should we exp ect for N n ? Unlik e the r ational case in the previo us subsection, the 6 vertex mo del with q 6 = 1 under the p erio dic b oundary condition do es not p ossess the global sl 2 -symmetry . Thus for the str ing solutions (13.1 0) to be complete, one should have N n = c n , where c n is the weigh t multiplicit y of the quantum space ( V ω 1 ) ⊗ L with weigh t ( L − 2 n ) ω 1 31 . Explicitly , c n =  L n  . This has bee n conﬁrmed for n = 1 , 2 in Ex ample 13.2. The next case is chec ked as N 3 = L N 3 =1 +     L − 2 2 2 L − 2     N 1 = N 2 =1 + L 1 3  L − 6 + 2 2  N 1 =3 = L ( L − 1)( L − 2) 6 . One ma y w onder what happ ens fo r n > L/ 2 where c n still ma kes s e ns e. The answer will b e given in the next s ubsection in a mor e genera l setting to g ether with the analogous r esult for b n . The only preliminary w e men tion here is that such consider ations ne c e ssarily in volve the situation P m < 0 hence the binomial co eﬃcients  X N  with X < N . 13.4. Theorems for type A 1 . W e hav e hitherto argued ab out three kinds of quantities (i) Number of string solutio ns in the B ethe ansatz, (ii) F er mio nic forms M n and N n , (iii) Representation theo r etical data b n and c n , 31 The same remark as the previous fo otnote applies here. 102 esp ecially without a muc h distinction b etw een (i) and (ii). Here we r edeﬁne (ii) without reco urse to (i) and formulate the theorems on the relations b etw ee n (ii) and (iii). W e tre at the g eneral spin case N m ≥ 1 ( V mω 1 ) ⊗ ν m and pre s ent the F ermionic character for m ulas. As p ow er series formulas, they are a ctually v alid for ar bitrary ν m ∈ C . The pro of of the theorem, which will b e outlined in the nex t subsection, do es not lea n on the string hypotheses but is solely der ived from the Q-sys tem. As such, it do es not prove nor disprov e the completeness of the s tr ing h yp o thesis. Let Q m ( Q m ) be the character (normalized character) of the irr educible m + 1 dimensional representation V mω 1 . Namely , Q m = χ ( V mω 1 ) = y m + y m − 2 + · · · + y − m = y m +1 − y − m − 1 y − y − 1 ( y = e ω 1 ) , (13.15) Q m = y − m Q m . (13.16) The Q m is a simpliﬁed notation fo r the v ar iable Q (1) m (13.1) in the Q-sys tem for A 1 : Q 2 m = Q m − 1 Q m +1 + 1 . (13.17) See (13.41). The Q m expressed as a function of Q 1 is the C hebyshev p olynomial o f the seco nd kind. In Section 13.5, w e will utilize the one adapted to the normalized character (13.16). Q m − 1 Q m +1 Q 2 m + y − 2 m Q − 2 m = 1 . (13 .18) Let ν m ∈ C ( m ∈ Z ≥ 1 ) be arbitra ry except that ν m = 0 for all but ﬁnitely many m . W e deﬁne the branching co eﬃcient b n and the w eig ht multiplicit y c n for all n ∈ Z ≥ 0 by Y m ≥ 1 ( Q m ) ν m = P n ≥ 0 b n y − 2 n 1 − y − 2 = X n ≥ 0 c n y − 2 n . (13.19) By the deﬁnition, the no rmalized character Q m is a p olynomia l in y − 2 with unit co n- stant term. ( Q m ) ν m denotes its ν m th pow er with unit constant term 1 + ν m ( Q m − 1) + ν m ( ν m − 1) 2 ( Q m − 1) 2 + · · · , w hich is a po lynomial or a p ow er series in y − 2 according as ν m ∈ Z ≥ 0 or not. When ν m ∈ Z ≥ 0 for a ny m ≥ 1, this deﬁni- tion of b n agrees with the one for the branching co eﬃcient o f V ( P m mν m − 2 n ) ω 1 in N m ≥ 1 ( V mω 1 ) ⊗ ν m for 0 ≤ n ≤ P m mν m / 2. The ab ove b n is a n extension o f this by b n = − b − n +1+ P m mν m , which is the skew symmetry under the W eyl gro up. As for the F er mionic forms, we r edeﬁne M n (13.8) and N n (13.14) by replacing P m (13.7) and the binomial co eﬃcient ther ein with the generalized ones 32 : P m = X k ≥ 1 min( m, k )( ν k − 2 N k ) , (13.20)  X N  = Q N i =1 ( X − i + 1) N ! ( X ∈ C , N ∈ Z ≥ 0 ) . (13.21) The sum ov er { N m | m ∈ Z ≥ 1 } is taken in the same w ay as (13.8) and (13.1 4). Namely , it is the ﬁnite sum over those N m ∈ Z ≥ 0 satisfying P m ≥ 1 mN m = n . There is no conditio n lik e P m ≥ 0 which do es no t make sense in the g eneral s etting 32 In Sections 13.2 and 13.3, the symbol  X N  wa s used only for 0 ≤ N ≤ X . 103 ν m ∈ C under consideration. The g eneralized binomial (13.21) is nonzero except the N p oints X = 0 , 1 , . . . , N − 1, and app ears in the expansio n (1 − x ) − β − 1 = ∞ X N =0  β + N N  x N , (13.2 2) for any β ∈ C . With these deﬁnitions we have Theorem 13.3 ([243, 245]) . The e qualities (1) M n = b n and (2) N n = c n hold for al l n ∈ Z ≥ 0 . Namely, the fol lowing p ower series formulas hold. Y m ≥ 1 ( Q m ) ν m = P n ≥ 0 M n y − 2 n 1 − y − 2 = X n ≥ 0 N n y − 2 n . (13.23) The formulas (1) and (2) a re due to [2 43] and [2 45], re s pe c tively . The theorem repro duces the obs erv ations in Sections 1 3 .2 a nd 13.3 in the sp ecia l cas e ν m = Lδ m 1 and 0 ≤ n ≤ L / 2, where P m ≥ 0 for any m ≥ 1 a utomatically ho lds. How ever, even for this simple choice ν m = Lδ m 1 , it further claims inﬁnitely many non tr ivial ident ities including M n = 0 for n ≥ L + 2 and N n = 0 and n ≥ L + 1. Example 13. 4 . Assume that ν m = 0 for m ≥ 4. Then LHS of (13.2 3) is (1 + y − 2 ) ν 1 (1 + y − 2 + y − 4 ) ν 2 (1 + y − 2 + y − 4 + y − 6 ) ν 3 . Setting γ m = P 3 k =1 min( m, k ) ν k , we write down M n (13.8) and N n (13.14) for n = 1 , 2 , 3. M 1 = γ 1 − 1 , M 2 = γ 2 − 3 + 1 2 ( γ 1 − 2)( γ 1 − 3) , M 3 = ( γ 3 − 5) + ( γ 1 − 3)( γ 2 − 5) + 1 6 ( γ 1 − 3)( γ 1 − 4)( γ 1 − 5) , N 1 = γ 1 , N 2 = γ 2 + 1 2 γ 1 ( γ 1 − 3) , N 3 = γ 3 +     γ 1 − 2 2 2 γ 2 − 2     + 1 6 γ 1 ( γ 1 − 4)( γ 1 − 5) . One can dir ectly chec k these co eﬃcients in the p ow er series expansio ns (13.23). F or instance in the simplest ca se ν m = 0 hence γ m = 0 for all m ≥ 1, all these co eﬃcients v anish except M 1 = − 1 as they should. In the case ν m ∈ Z ≥ 0 ( m ≥ 1), P m in (13.20) can b e a no nnegative integer for some { N m } . Then it makes sense to in tro duce the following v ariant of M n : M n = X { N m } + Y m ≥ 1  P m + N m N m  , (13.24 ) where P m and  X N  are again sp eciﬁed by (13.20) and (13.21) a s for M n . The only diﬀerence from it is that the sum P + { N m } extends over those N m ∈ Z ≥ 0 satisfying n = P m ≥ 1 mN m with the extra condition P m ≥ 0 if N m ≥ 1. Given { ν m } , n and { N m } satisfying P m ≥ 1 mN m = n , let m 0 be the maximal m such that N m ≥ 1. Then we hav e P m 0 = P k ≥ 1 min( m 0 , k ) ν k − 2 n ≤ P k ≥ 1 k ν k − 2 n . Thu s w e see M n = 0 if n > 1 2 P k ≥ 1 k ν k . Theorem 13.5 ([24 7, 2 48]) . F or any ν m ∈ Z ≥ 0 , the e quality M n = b n holds for 0 ≤ n ≤ 1 2 P m ≥ 1 mν m . 104 As remar ked after Theor em 13 .3, Theor em 13 .5 is eq uiv ale n t to Theorem 13 .3 (1) in the the sp ecial cas e ν m = Lδ m 1 and 0 ≤ n ≤ L/ 2 . In g e neral, they imply that the cont ributions to N n inv olving P m < 0 cancel out. Example 13.6. T ak e ν m = 2 δ m 3 in E xample 13 .4. Then ( γ 1 , γ 2 , γ 3 ) = (2 , 4 , 6). The three terms in M 3 corres p o nd to choos ing nonzero N m as N 3 = 1, N 1 = N 2 = 1 and N 1 = 3. The relev ant P m ’s ar e P 3 = 0, P 1 = P 2 = − 2 and P 1 = − 4 , resp ectively . Thus M 3 is given by the ﬁrst term only γ 3 − 5 = 1 . This co incides with M 3 since the other tw o terms cancel. 13.5. Multiv ariable Lagrange in version. Her e we outline the pro o f of Theor em 13.3. W e desc r ib e a n essential step of de r iving (13 .23) from (13.18) in a gener alized setting applicable to g case [249]. Let H denote a ﬁnite index set. Let w = ( w i ) i ∈ H and v = ( v i ) i ∈ H be complex m ultiv ariables, a nd let G = ( G ij ) i,j ∈ H be a complex square matrix of size | H | . W e consider a holomorphic map D → C H , v 7→ w ( v ) with w i ( v ) = v i Y j ∈ H (1 − v j ) − G ij , (13.25) where D is some neighbor ho o d of v = 0 in C H . The Jaco bian ( ∂ w/∂ v )( v ) is 1 a t v = 0, so that the map w ( v ) is bijective ar ound v = w = 0. Let v ( w ) be the in verse map around v = w = 0. Inv erting (13.2 5), we o btain the following functional equation for v i ( w )’s: v i ( w ) = w i Y j ∈ H (1 − v j ( w )) G ij . (13 .26) By in tro ducing new functions Q i ( w ) = 1 − v i ( w ) , (13.27) the equation (13.26) is written as Q i ( w ) + w i Y j ∈ H Q j ( w ) G ij = 1 . (13.28 ) F rom now on, we rega rd (13.28) a s equa tions for a family ( Q i ( w )) i ∈ H of p ow e r series of w = ( w i ) i ∈ H with the unit constant ter ms. The pro cedur e fro m (13.25) to (13.28) can b e reversed; therefore, the p ow er series expa nsion of Q i ( w ) in (13.27) gives the unique family ( Q i ( w )) i ∈ H of p ow er ser ies of w with the unit constant terms which satisﬁes (13.2 8). W e deﬁne (ﬁnite) Q-system to b e the following equations for a family ( Q i ( w )) i ∈ H of power serie s o f w with the unit constant terms: Y j ∈ H Q j ( w ) D ij + w i Y j ∈ H Q j ( w ) G ij = 1 ( i ∈ H ) , (13.29) where D = ( D ij ) i,j ∈ H and G = ( G ij ) i,j ∈ H are arbitrar y complex ma trices with det D 6 = 0. E quation (13 .28), which is the spec ial case of (1 3.29) with D = I ( I : the identit y matrix), is ca lled a standard Q -system. By setting Q ′ i ( w ) = Q j ∈ H Q j ( w ) D ij , (1 3.29) is alwa ys transformed to the standar d one (13.28) with G replaced b y G ′ = GD − 1 and vice v ers a. Therefor e , the Q-system (13.29) a lso has the unique solution. 105 Given the Q- system (13.29) and ν = ( ν i ) i ∈ H ∈ C H , we deﬁne tw o p ow er serie s of w M ν ( w ) = X N M ( ν, N ) w N , N ν ( w ) = X N N ( ν, N ) w N , (1 3 .30) where w N = Q i ∈ H w N i i and the sums run ov er N = ( N i ) i ∈ H ∈ ( Z ≥ 0 ) H . The co eﬃcients are given by M ( ν, N ) = Y i ∈ H ( N )  P i + N i N i  , (13.31) N ( ν, N ) =  det H ( N ) F ij  Y i ∈ H ( N ) 1 N i  P i + N i − 1 N i − 1  , (13.32) where the binomial is deﬁned b y (13 .21) and w e have set H ( N ) = { i ∈ H | N i 6 = 0 } , P i = P i ( ν, N ) := − X j ∈ H ν j ( D − 1 ) j i − X j ∈ H N j ( GD − 1 ) j i , (13 .33) F ij = F ij ( ν, N ) := δ ij P j + ( GD − 1 ) ij N j . ( 13.34) det H ( N ) is a shorthand notation for det i,j ∈ H ( N ) . In (13.3 1) and (1 3.32), det ∅ and Q ∅ mean 1; therefor e , M ν ( w ) and N ν ( w ) are power series w ith the unit constant terms. See [2 49, section 2] for the convergence radius. No te a similarity to (13 .8) and (13.14). Theorem 13 .7 ([2 4 9]) . L et ( Q i ( w )) i ∈ H b e the unique solution of (13.29). F or ν = ( ν i ) i ∈ H ∈ C H , the fol lowing formulas ar e valid: Y i ∈ H Q i ( w ) ν i = M ν ( w ) M 0 ( w ) = N ν ( w ) . (13.35 ) Q i ( w ) itself is obtained by s e tting ν j = δ ij . Example 13.8. Let | H | = 1 . Then, (13.29) is a n equation for a single p ower s eries Q ( w ): Q ( w ) D + w Q ( w ) G = 1 , where D 6 = 0 a nd G are complex nu mbers and Theorem 13.7 shows that Q ( w ) ν = N ν ( w ) = ν D ∞ X N =0 Γ (( ν + N G ) /D )( − w ) N Γ (( ν + N G ) /D − N + 1) N ! . This p ow er s eries formula is well known and have a very long histor y since Lamber t (e.g. [250, pp. 306–3 07]). As noted b e fo re, the Q-system (13.29) is bijectiv ely transfo r med to the standard one (1 3 .28). Under the corres po nding c hanges D → I , ν i → P j ∈ H ν j ( D − 1 ) j i and G → GD − 1 , quan tities (13.33) and (13.3 4) remain inv aria nt , hence so are M ( ν, N ) and M ( ν , N ). Thus we hav e only to prove Theore m 13.7 for the s ta ndard case D = I , where Q i ( w ) is descr ib ed by (13.2 5)–(13.27). Therefor e, Theorem 13.7 follows from 106 Prop ositio n 13.9 ([24 9] Prop osition 2.8) . L et v = v ( w ) b e the inverse map of (13.25). Le t M ν ( w ) and N ν ( w ) b e those for D = I in (13.33) and (13.34). Then, the p ower series exp ansions det H  w j v i ∂ v i ∂ w j ( w )  Y i ∈ H (1 − v i ( w )) ν i − 1 = M ν ( w ) , (13.3 6) Y i ∈ H (1 − v i ( w )) ν i = N ν ( w ) (13.37) hold ar oun d w = 0 . This is a particular ly nice example of the multiv a riable Lagra nge inv ersio n for- m ula (e.g. [251]), where all the calculations can be carr ie d through by a multiv a r i- able residue analysis. Pr o of. The ﬁrst formula (13 .36). W e ev aluate the co eﬃcient for w N on the LHS of (13.36) as follows: Res w =0 ∂ v ∂ w ( w ) Y i ∈ H n (1 − v i ( w )) ν i − 1 ( v i ( w )) − 1 ( w i ) 1 − N i − 1 o dw = Res v =0 Y i ∈ H n (1 − v i ) ν i − 1 ( v i ) − 1  v i Y j ∈ H (1 − v j ) − G ij  − N i o dv = Res v =0 Y i ∈ H n (1 − v i ) − P i ( ν,N ) − 1 ( v i ) − N i − 1 o dv = Y i ∈ H  P i ( ν, N ) + N i N i  = M ( ν, N ) , where we used (13.22) to get the la st line. Thus, (13.3 6) is prov ed. The se c ond formula (13.37). By a simple calculation, we hav e det H  v j w i ∂ w i ∂ v j ( v )  Y i ∈ H (1 − v i ) = det H  δ ij + ( − δ ij + G ij ) v i  = X J ⊂ H d J Y i ∈ J v i , (13.3 8 ) where d J := det J ( − δ ij + G ij ), and the sum is taken ov er a ll the subsets J of H . Therefore, the LHS of (13.37) is written as det H  w j v i ∂ v i ∂ w j ( w )  X J ⊂ H d J Y i ∈ H n (1 − v i ( w )) ν i − 1 v i ( w ) θ ( i ∈ J ) o . ( 13.39) By a simila r residue calcula tio n as ab ove, the co eﬃcient for w N of (13 .39) is ev al- uated as ( θ (true ) = 1 and θ (false ) = 0) X J ⊂ H d J Res v =0 Y i ∈ H n (1 − v i ) − P i ( ν,N ) − 1 ( v i ) − N i + θ ( i ∈ J ) − 1 o dv = X J ⊂ H ( N ) d J Y i ∈ H ( N )  P i ( ν, N ) + N i − θ ( i ∈ J ) N i − θ ( i ∈ J )  =  X J ⊂ H ( N ) d J Y i ∈ J N i Y i ∈ H ( N ) \ J ( P i + N i )  Y i ∈ H ( N ) 1 N i  P i + N i − 1 N i − 1  = det H ( N )  δ ij ( P j + N j ) + ( − δ ij + G ij ) N j  Y i ∈ H ( N ) 1 N i  P i + N i − 1 N i − 1  = N ( ν , N ) . 107 This co mpletes the pro of o f Theore m 13.7. What is left to prove Theorem 1 3.3 from it? Co mpa ring the Q-systems (13 .29) and (13.18) and also P m in (13 .3 3) and (13.20), we see that Theorem 13 .3 for mally corres po nds to taking H = Z ≥ 1 , w i = y − 2 i , ( D − 1 ) ij = − min( i, j ) , D ij = δ i,j +1 + δ i,j − 1 − 2 δ ij , G ij = − 2 δ ij (13.40) in Theo rem 13 .7, and c la iming M 0 ( w ) = 1 − y − 2 thereunder. Since we s tarted with the a ssumption that H is a ﬁnite set, it is nontrivial how to make sense of these choices and claims. W e r efer to [249] for a proper treatment o f such an inﬁnite ( | H | = ∞ ) Q-system as a pro jective limit of the ﬁnite Q-systems. According a result therein, Theor em 13 .3 is shown, among other things , from the conv erg ence prop erty: the limit lim m →∞ Q m ( w i = y − 2 i ) exists in C [[ y − 2 ]]. 13.6. Q-system and theorems for g . Here we pr esent the Q-system and analo g of Theorem 13.3 a nd Theor em 13.5 for gener al g . W e use the notations in Section 2.1 such a s I , t , t a , C = ( C ab ), α a and ω a . The unrestricted Q -system for g is the following relatio ns a mong the v ar iables { Q ( a ) m | a ∈ I , m ≥ 1 } , where Q (0) m = Q ( a ) 0 = 1 if they o ccur on the RHS. F or simply lac e d g , ( Q ( a ) m ) 2 = Q ( a ) m − 1 Q ( a ) m +1 + Y b ∈ I : C ab = − 1 Q ( b ) m . (13.4 1) F or g = B r , ( Q ( a ) m ) 2 = Q ( a ) m − 1 Q ( a ) m +1 + Q ( a − 1) m Q ( a +1) m (1 ≤ a ≤ r − 2 ) , ( Q ( r − 1) m ) 2 = Q ( r − 1) m − 1 Q ( r − 1) m +1 + Q ( r − 2) m Q ( r ) 2 m , ( Q ( r ) 2 m ) 2 = Q ( r ) 2 m − 1 Q ( r ) 2 m +1 + ( Q ( r − 1) m ) 2 , ( Q ( r ) 2 m +1 ) 2 = Q ( r ) 2 m Q ( r ) 2 m +2 + Q ( r − 1) m Q ( r − 1) m +1 . (13.42) F or g = C r , ( Q ( a ) m ) 2 = Q ( a ) m − 1 Q ( a ) m +1 + Q ( a − 1) m Q ( a +1) m (1 ≤ a ≤ r − 2 ) , ( Q ( r − 1) 2 m ) 2 = Q ( r − 1) 2 m − 1 Q ( r − 1) 2 m +1 + Q ( r − 2) 2 m ( Q ( r ) m ) 2 , ( Q ( r − 1) 2 m +1 ) 2 = Q ( r − 1) 2 m Q ( r − 1) 2 m +2 + Q ( r − 2) 2 m +1 Q ( r ) m Q ( r ) m +1 , ( Q ( r ) m ) 2 = Q ( r ) m − 1 Q ( r ) m +1 + Q ( r − 1) 2 m . (13.43) F or g = F 4 , ( Q (1) m ) 2 = Q (1) m − 1 Q (1) m +1 + Q (2) m , ( Q (2) m ) 2 = Q (2) m − 1 Q (2) m +1 + Q (1) m Q (3) 2 m , ( Q (3) 2 m ) 2 = Q (3) 2 m − 1 Q (3) 2 m +1 + ( Q (2) m ) 2 Q (4) 2 m , ( Q (3) 2 m +1 ) 2 = Q (3) 2 m Q (3) 2 m +2 + Q (2) m Q (2) m +1 Q (4) 2 m +1 , ( Q (4) m ) 2 = Q (4) m − 1 Q (4) m +1 + Q (3) m . 108 F or g = G 2 , ( Q (1) m ) 2 = Q (1) m − 1 Q (1) m +1 + Q (2) 3 m , ( Q (2) 3 m ) 2 = Q (2) 3 m − 1 Q (2) 3 m +1 + ( Q (1) m ) 3 , ( Q (2) 3 m +1 ) 2 = Q (2) 3 m Q (2) 3 m +2 + ( Q (1) m ) 2 Q (1) m +1 , ( Q (2) 3 m +2 ) 2 = Q (2) 3 m +1 Q (2) 3 m +3 + Q (1) m ( Q (1) m +1 ) 2 . (13.44) These relations are uniformly written as ( Q ( a ) m ) 2 = Q ( a ) m − 1 Q ( a ) m +1 + ( Q ( a ) m ) 2 Y ( b,k ) ∈ H ( Q ( b ) k ) G am,bk , (1 3.45) by using the no tations (13 .48) and (1 3 .51). W e shall introduce the r estricted Q - system in Section 14.5. As men tioned aro und (13.1), thes e relations follow fr o m the T- systems b y for- getting the sp ectra l par ameter u . Recall tha t res χ q ( W ( a ) m ( u )) denotes the cla s sical character o f the Kirillov-Reshetikhin mo dule W ( a ) m ( u ). See (4.2 3) for the deﬁnition of r es. Since res r emov es the dep endence on u , w e will simply write as res χ q ( W ( a ) m ) in what follows. The follo wing is a corolla ry of Theore m 4.8. Prop ositio n 13. 10. The substitu tion Q ( a ) m = res χ q ( W ( a ) m ) s atisﬁ es the unr estricte d Q-system. F rom now on, we under stand the symbol Q ( a ) m as repr esenting res χ q ( W ( a ) m ). By Theorem 4.6 (1), the normalized character Q ( a ) m = e − mω a Q ( a ) m (13.46) is a po lynomial in e − α 1 , . . . , e − α r with unit constant term a nd co eﬃcients from Z ≥ 0 . In terms of Q ( a ) m , the Q-system is expressed as Y ( b,k ) ∈ H ( Q ( b ) k ) D am,bk + e − mα a Y ( b,k ) ∈ H ( Q ( b ) k ) G am,bk = 1 (13.47) for ( a, m ) ∈ H . Here H , D am,bk and G am,bk are deﬁned by H = { ( a, m ) | a ∈ I , m ∈ Z ≥ 1 } , (13 .48) D am,bk = − δ ab (2 δ mk − δ m,k +1 − δ m,k − 1 ) , (13.49) ( D − 1 ) am,bk = − δ ab min( m, k ) . (13.50) G am,bk =          − C ba ( δ m, 2 k − 1 + 2 δ m, 2 k + δ m, 2 k +1 ) t a /t b = 2 , − C ba ( δ m, 3 k − 2 + 2 δ m, 3 k − 1 + 3 δ m, 3 k t a /t b = 3 , +2 δ m, 3 k +1 + δ m, 3 k +2 ) − C ab δ t b m,t a k otherwise . (13.51) F or g = A 1 , the data H, D , G here r educe to (13.4 0) hence (13.47) to (13.1 8). By a n a nalysis par allel with A 1 case, one can establish the power series formulas inv olving F er mionic for ms. They are read oﬀ (13.30)–(13.34) b y formally replacing the single indices by double ones as i → ( a, m ), j → ( b, k ), etc. T o be concrete, let ν = ( ν ( a ) m ) ( a,m ) ∈ H ∈ C H , wher e ν ( a ) m = 0 fo r all but ﬁnit ely many ( a, m ). F or 109 N = ( N ( a ) m ) ( a,m ) ∈ H ∈ ( Z ≥ 0 ) H , we deﬁne M ( ν, N ) = Y ( a,m ) ∈ H ( N )  P ( a ) m + N ( a ) m N ( a ) m  , (13.52) N ( ν, N ) =  det H ( N ) F am,bk  Y ( a,m ) ∈ H ( N ) 1 N ( a ) m  P ( a ) m + N ( a ) m − 1 N ( a ) m − 1  , (1 3.53) where the binomial is the generalize d one (13.2 1). W e hav e se t H ( N ) = { ( a, m ) ∈ H | N ( a ) m 6 = 0 } and det H ( N ) denotes det ( a,m ) , ( b,k ) ∈ H ( N ) . Deﬁne fur ther P ( a ) m = X k ≥ 1 min( m, k ) ν ( a ) k − X ( b,k ) ∈ H ( α a | α b ) min( t b m, t a k ) N ( b ) k , (13.54) F am,bk = δ ab δ mk P ( a ) m + ( α a | α b ) min( t b m, t a k ) N ( b ) k . (13.55) With these deﬁnitions we have Theorem 13.11 ([81, 252, 8 0, 2 49, 68]) . The fol lowing p ower s eries formulas ar e valid: Y ( a,m ) ∈ H ( Q ( a ) m ) ν ( a ) m = P N M ( ν, N ) e − P ( a,m ) ∈ H mN ( a ) m α a Q α ∈ ∆ + (1 − e − α ) = X N N ( ν, N ) e − P ( a,m ) ∈ H mN ( a ) m α a , (13.56) wher e the sums run over N = ( N ( a ) m ) ( a,m ) ∈ H ∈ ( Z ≥ 0 ) H without any c onstr aints. The symb ol ∆ + denotes t he set of p ositive ro ots of g . See Section 13.8 how this theorem was established by in teg r ating man y works. Let us turn to the sp ecia l ca s e ν ( a ) m ∈ Z ≥ 0 for any ( a, m ) ∈ H . Then the p ow er series (13.56) actually trunca tes to a p olyno mia l, and Theor em 13.11 implies the F ermionic formulas for the branching co eﬃcient b λ and the weigh t multiplicit y c λ in (13.2). T o wr ite them do wn, we introduce M λ = X N M ( ν, N ) , N λ = X N N ( ν, N ) ( λ ∈ r X a =1 Z ω a ) , (13 .5 7) where the sums run ov er N = ( N ( a ) m ) ( a,m ) ∈ H ∈ ( Z ≥ 0 ) H satisfying the weight con- dition λ = X ( a,m ) ∈ H mν ( a ) m ω a − X ( a,m ) ∈ H mN ( a ) m α a . (1 3.58) Then the following is a co rollary of Theorem 13.11: Y a,m ( Q ( a ) m ) ν ( a ) m = X λ b λ χ ( V λ ) , b λ = M λ for λ ∈ r X a =1 Z ≥ 0 ω a , (13.59) Y a,m ( Q ( a ) m ) ν ( a ) m = X λ c λ e λ , c λ = N λ for λ ∈ r X a =1 Z ω a . (13.60 ) As the generalizatio n of (13.24), we further int ro duce M λ = X N + M ( ν, N ) , (13.61) 110 where the sum P + N extends over N = ( N ( a ) m ) ( a,m ) ∈ H ∈ ( Z ≥ 0 ) H satisfying (13.58) and the extra condition that P m ≥ 0 w he never N m ≥ 1. The n the following is the g v ersion of Theorem 13.5. Theorem 13 .12 ([253, 247, 248, 254]) . F or λ ∈ P r a =1 Z ≥ 0 ω a , the e quality b λ = M λ is valid. 13.7. Q ( a ) m as a classi cal c haracter. Her e w e pres ent the ex pansion of Q ( a ) m int o classical characters. Suc h an example has already b een given in (4.24) for the rank 2 algebras g = A 2 , B 2 , C 2 and G 2 . Here are a few examples from E 8 : Q (1) 1 = χ ( V ω 1 ) + χ ( V 0 ) , Q (1) 2 = χ ( V 2 ω 1 ) + χ ( V ω 1 ) + χ ( V 0 ) , Q (2) 1 = χ ( V ω 2 ) + 2 χ ( V ω 1 ) + χ ( V ω 7 ) + χ ( V 0 ) , Q (3) 1 = χ ( V ω 3 ) + 2 χ ( V ω 8 ) + 4 χ ( V ω 7 ) + χ ( V ω 1 + ω 7 ) + 3 χ ( V ω 2 ) + χ ( V 2 ω 1 ) + 4 χ ( V ω 1 ) + 2 χ ( V 0 ) , which satisfy a Q-system relation ( Q (1) 1 ) 2 = Q (1) 2 + Q (2) 1 for instance . In gener al from (13.59) and (13.58), the expansion takes the for m Q ( a ) m = χ ( V mω a ) + called “childre n” z }| { X λ 0, ǫ = ± 1 and ( p, s ) ∈ H ℓ are mo del par a meters sp ecifying the temp er- ature, nor malization of energ y , tw o critica l regimes and repre sentation W ( p ) s (fusion t yp e ) with which the mo del is a sso ciated, resp ectively . The physical meaning of ǫ ( a ) m ( u ) is the pseudo energy deﬁned by exp( − β ǫ ( a ) m ( u )) = ρ ( a ) m ( u ) /σ ( a ) m ( u ) in terms of the co lo r a length m string density ρ ( a ) m ( u ) and hole density σ ( a ) m ( u ). More details can be found in Section 15.1, but we do not ne e d those background here. W e assume tha t (14.5) can be ana ly tically contin ued oﬀ the rea l axis o f u until |ℑ m u | ≤ 1 . Setting u → u ± i ∓ 0 i , take the sum of the resulting tw o equations. The LHS v anishes and the RHS is ev aluated by means of 1 4 co sh π 2 ( u − v + i − 0 i ) + 1 4 co sh π 2 ( u − v − i + 0 i ) = δ ( u − v ) (14.6) as the c o nv olution kernel. By intro ducing the v ariable Y ( a ) m ( u ) = exp( − β ǫ ( a ) m ( u )), the Boltzmann factor of the pseudo energy , the r esult is the logar ithm of Y ( a ) m ( u − i ) Y ( a ) m ( u + i ) = Q b ∈ I (1 + Y ( b ) m ( u )) N ab (1 + Y ( a ) m − 1 ( u ) − 1 )(1 + Y ( a ) m +1 ( u ) − 1 ) . (14.7) This is the Y-system for g = A r , D r and E 6 , 7 , 8 (2.11) in the conven tion that Y ( a ) m ( u + k ) there b ecomes Y ( a ) m ( u + ik ). It is le vel ℓ res tricted since only Y ( a ) m ( u ) with ( a, m ) ∈ H ℓ are present. Notice tha t the LHS of (14.5) that had carr ied the mo del dep endent information β , γ , ǫ and ( p, s ) disappea red all tog e ther. In this sense, the Y-system is a universal feature of all the ph ysical systems describ ed by the TBA equation (14.5) whose LHS is a ny 2 i -antiperio dic function of u . Put it diﬀerently , the LHS enco des the sp eciﬁc prop er ties in each mo del that are co upled as a driving term to the universal structure (Y-system). Let us obser ve another a sp ect of the Y-system (14.7). It is wr itten as (1 + Y ( a ) m ( u − i ) − 1 )(1 + Y ( a ) m ( u + i ) − 1 ) (1 + Y ( a ) m − 1 ( u ) − 1 )(1 + Y ( a ) m +1 ( u ) − 1 ) = (1 + Y ( a ) m ( u − i ))(1 + Y ( a ) m ( u + i )) Q b ∈ I (1 + Y ( b ) m ( u )) N ab . (14.8) The LHS and RHS o f (1 4.7) p osses s parallel structur es related to A ℓ − 1 and g , resp ectively . In the F ourie r spa ce they a re enco ded in the defor med Cartan matrices with indices corres p o nding to the length m and the color a , resp ectively . T o s e e it, deﬁne the F ourier transfo rmation ˆ f = ˆ f ( x ) of f = f ( u ) by f ( u ) = 1 2 π Z ∞ −∞ ˆ f ( x ) e iux dx, ˆ f ( x ) = Z ∞ −∞ f ( u ) e − iux du. (14.9) If we for ma lly interpret the multiplication with e cx ( c ∈ R ) in the F our ier ( x ) space as the diﬀerence op era tor u → u − ic in the “r eal” ( u ) s pa ce, the lo garithm of the RHS o f (1 4 .8) is a ssigned with the F o urier co mp onent P b ∈ I ˆ M ab ( x ) b ln  1 + Y ( b ) m  , 114 where ˆ M ab ( x ) = 2 δ ab cosh x − N ab (for ADE) (14.10) is the deformed Cartan matr ix of g . Actually the F ourier transformation of the TBA equa tion (14 .5) co nt ains P b ∈ I ˆ M ab ( x ) 2 cosh x b ln  1 + Y ( b ) m  so that the identit y (14.6) works in the real space. Paralle l remark s a pply to the LHS of (1 4 .8). W e ca ll the functions like ˆ M ab ( x ) TBA kernels as they emerge in the TBA calculation (Section 15.1) and pla y important roles as building blo cks of integral kernels in the TBA equa tion. 14.2. TBA k ernels. Her e we summarize the deﬁnitions and useful prop erties o f the TB A kernels for general g . In place o f (14 .10), we redeﬁne ˆ M ab ( x ) and intro duce ˆ K mn a ( x ) as ˆ M ab ( x ) = 2 δ ab cosh  x t a  − N ab = B ab + 2 δ ab  cosh  x t a  − 1  , (14.1 1) ˆ K mn a ( x ) = δ mn − δ m,n − 1 + δ m,n +1 2 co sh  x t a  . (14.12) F or ( a, m ) , ( b, k ) ∈ H ℓ , we further intro duce ˆ A mk ab ( x ) = sinh  min( m t a , k t b ) x  sinh  ( ℓ − max( m t a , k t b )) x  sinh( x t ab ) sinh( ℓx ) , (14.13) ˆ K mk ab ( x ) = ˆ A mk ab ( x ) ˆ M ab ( x ) , (14.14) ˆ J mk ab ( x ) = ℓ a − 1 X n =1 ˆ K mn a ( x ) ˆ K nk ab ( x ) = ˆ M ab ( x ) ˆ P mk ab ( x ) 2 co sh  x t a  , (14.15) ˆ P mk ab ( x ) = 2 co sh  x t a  ℓ a − 1 X n =1 ˆ K mn a ( x ) ˆ A nk ab ( x ) (14.16) = sinh( x t a ) sinh( x t ab ) δ t b m,t a k + t b − t a X j =1 sinh( j x t b ) sinh( x t b )  δ t b t a ( m +1) − j, k + δ t b t a ( m − 1)+ j, k  . The sum P t b − t a j =1 in (14.16) is to b e understo o d as zero if t a ≥ t b . Since the la tter expressions in (14.15) and (14.16) do not contain ℓ , we can and do extend the deﬁnition o f ˆ J mk ab ( x ) a nd ˆ P mk ab ( x ) to all the nonneg a tive integers m, k ≥ 0. The inv erse F our ier trans form J mk ab ( u ) is a n even function of u but J mk ab ( u ) 6 = J km ba ( u ) in general as o ppo sed to ˆ A mk ab ( x ) = ˆ A km ba ( x ) and ˆ K mk ab ( x ) = ˆ K km ba ( x ). The ˆ K mn a ( x ) in (14 .1 2) should be distinguished from ˆ K mn aa ( x ) in (14.14). The following r elations are easily chec ked: 2 co sh  x t a  ℓ a − 1 X n =1 ˆ A mn aa ( x ) ˆ K nk a ( x ) = δ mk , (14.17) 2 co sh  x t a  ℓ a − 1 X n =1 ˆ A mn aa ( x ) ˆ J nk ab ( x ) = ˆ K mk ab ( x ) , (14.1 8) ˆ J mk ab ( x ) = δ ab δ mk − N ab ˆ P mk ab ( x ) 2 co sh  x t a  . (14.1 9) 115 All the TBA kernels (14.11)–(14.16) ar e deduced fro m ˆ A mn ab ( x ) and ˆ M ab ( x ) by using these relations. The bas ic ones ˆ A mn ab ( x ) and ˆ M ab ( x ) are obtained as Z ∞ −∞ due − iux ∂ ∂ u Θ m a  u, s t a  = ˆ A ms aa ( x ) | ℓ → L , (14.20) Z ∞ −∞ due − iux ∂ ∂ u Θ mk ab ( u, ( α a | α b )) = − δ ab δ mk + ˆ M ab ( x ) ˆ A mk ab ( x ) | ℓ → L , (14.21) where Θ m a  u, s t a  (15.3) and Θ mk ab ( u, ( α a | α b )) (15.4) a r e the log arithm o f the LHS and the RHS of the Bethe equatio n under the string hypothesis , resp ectively . Se e (15.1)–(15.4). When g is simply laced, the TBA kernels simplify as ˆ A mk ab ( x ) = sinh  min( m, k ) x  sinh  ( ℓ − max( m, k )) x  sinh x sinh ( ℓx ) , (14.22) ˆ J mk ab ( x ) = ˆ M ab ( x ) δ mk 2 co sh x =  δ ab − N ab 2 co sh x  δ mk , (14.2 3) ˆ P mk ab ( x ) = δ mk . (14.24) 14.3. Y-system for g from TBA e quation. Let us derive the level ℓ r estricted Y-system for general g from the TBA equa tion. W e q uote the latter obtained in (15.13) with the notation Y ( a ) m ( u ) = e xp( − β ǫ ( a ) m ( u )): ǫβ γ δ pa δ sm 4 t − 1 p cosh( t p π u/ 2) = − ln Y ( a ) m ( u ) − Z ∞ −∞ dv ln h  1 + Y ( a ) m − 1 ( v ) − 1  1 + Y ( a ) m +1 ( v ) − 1  i 4 t − 1 a cosh( t a π ( u − v ) / 2) + X ( b,k ) ∈ H ℓ N ab Z ∞ −∞ dv h P mk ab ∗ ln  1 + Y ( b ) k  i ( v ) 4 t − 1 a cosh( t a π ( u − v ) / 2) . (14.25) P mk ab is deﬁned via its F ourier comp onent (14.16) and ∗ denotes the conv olution ( f 1 ∗ f 2 )( u ) = Z ∞ −∞ dv f 1 ( u − v ) f 2 ( v ) . (14 .26) As the simply laced case, we assume that (14 .25) ca n be a na lytically con tin ued o ﬀ the r eal axis o f u until |ℑ m u | ≤ t − 1 a . Then the sum after the shifts u → u ± t − 1 a i ∓ 0 i eliminates the LHS, giving ln h Y ( a ) m ( u − i t a ) Y ( a ) m ( u + i t a ) i = − ln h  1 + Y ( a ) m − 1 ( u ) − 1  1 + Y ( a ) m +1 ( u ) − 1  i + X ( b,k ) ∈ H ℓ N ab h P mk ab ∗ ln  1 + Y ( b ) k  i ( u ) . (14.27) F or simply laced alg ebras, P mk ab ( u ) = δ mk δ ( u ) by (14.24), and we ar e done. T o illustrate the genera l case, take g = G 2 with ( a, b ) = (1 , 2) as an example. Then 116 ( t a , t b ) = (1 , 3) and (14 .16) reads ˆ P mk ab ( x ) = ˆ P mk 12 ( x ) = ( e 2 x 3 + 1 + e − 2 x 3 ) δ 3 m,k + δ 3 m − 2 ,k + δ 3 m +2 ,k + ( e x 3 + e − x 3 )( δ 3 m − 1 ,k + δ 3 m +1 ,k ) , P mk 12 ( u ) = ( δ ( u − 2 i 3 ) + δ ( u ) + δ ( u + 2 i 3 )) δ 3 m,k + δ ( u )( δ 3 m − 2 ,k + δ 3 m +2 ,k ) + ( δ ( u − i 3 ) + δ ( u + i 3 ))( δ 3 m − 1 ,k + δ 3 m +1 ,k ) . If ln(1 + Y (2) k ( v )) is analy tic in the s trip |ℑ m v | ≤ 2 3 35 and decays rapidly as |ℜ e v | → ∞ , one can shift the c onv olution in tegra l R dv P mk 12 ( u − v ) ln(1 + Y (2) k ( v )) oﬀ the real ax is of v to pick the supp or t of delta functions. In this wa y the last ter m in (14.27) gives the loga r ithm of (1 + Y (2) 3 m  u − 2 i 3  )(1 + Y (2) 3 m ( u ))(1 + Y (2) 3 m  u + 2 i 3  )(1 + Y (2) 3 m − 2 ( u ))(1 + Y (2) 3 m +2 ( u )) × (1 + Y (2) 3 m − 1  u − i 3  )(1 + Y (2) 3 m − 1  u + i 3  )(1 + Y (2) 3 m +1  u − i 3  )(1 + Y (2) 3 m +1  u + i 3  ) . This is the numerator of the RHS in the ﬁrst rela tion of the Y-system for G 2 (2.15) with the shift unit mult iplied by i . The gene r al case is similar and (14.27) gives rise to the logar ithmic form of the (restricted) Y-sys tem for g . On account of (1 4.16), in gener a l it suﬃces to as s ume that ln(1 + Y ( a ) m ( u )) is analytic in the strip |ℑ m u | ≤ t a − 1 t a and decays rapidly as |ℜ e u | → ∞ . If the analyticity arg ument can b e left out, the Y-system is deduced more quickly from the TBA kernels in the F ourie r spa c e . In fact, one ca n star t w ith the TBA equation (15.12) without the LHS 36 : ℓ a − 1 X n =1 ˆ K mn a ( x ) b ln  1 + ( Y ( a ) n ) − 1  = X ( b,k ) ∈ H ℓ ˆ J mk ab ( x ) b ln  1 + Y ( b ) k  . (14.28) Multiply with 2 cosh( x t a ) and use (14.12) and (14.1 9) to rearr ange it slightly a s 2 co sh  x t a  b ln Y ( a ) m = X ( b,k ) ∈ H ℓ N ab ˆ P mk ab ( x ) b ln  1 + Y ( b ) k  − b ln h  1 + ( Y ( a ) m − 1 ) − 1  1 + ( Y ( a ) m +1 ) − 1  i . (14.29) This is the Y-system if cosh( x t a ) and ˆ P mk ab ( x ) (14.16) ar e rega rded as the diﬀerence op erator s as men tio ned after (14.9). W e hav e demonstrated that the Y-system is a diﬀere nc e equa tion whose structur e is governed by the TBA kernels. On the other hand, recall that Theorem 2 .5 oﬀers another ro ute to o btain the Y-system by inv o king its connectio n to the T-sy stem. It is y et to be understo o d why the tw o “characteriz a tions” of the Y-sys tem coincide. 14.4. Constan t Y-system. In either unres tricted or level ℓ res tr icted Y-system, one can discard the dep e ndence of Y ( a ) m ( u ) on u . The re sulting algebr a ic equa tion on Y ( a ) m = Y ( a ) m ( u ) is called the unrestricted or level ℓ r estricted constant Y-sy s tem 37 . 35 Actually |ℑ m v | ≤ 2 3 for ln(1 + Y (2) 3 m ( v )) and |ℑ m v | ≤ 1 3 for ln(1 + Y (2) 3 m ± 1 ( v )) suﬃce. 36 According to our previous argument, it i s actually more prop er to suppress the LHS after multiplying 2 cosh( x t a ). 37 The lev el ℓ restricted constan t Y-system here is the same with the one introduced in Section 5.1. 117 The unrestricted consta n t Y-system for g is the set of algebraic equations on { Y ( a ) m | ( a, m ) ∈ H } . ( H is deﬁned in (13.48).) F or simply laced g , it has the for m ( Y ( a ) m ) 2 = Q b ∈ I : C ab = − 1 (1 + Y ( b ) m ) (1 + ( Y ( a ) m − 1 ) − 1 )(1 + ( Y ( a ) m +1 ) − 1 ) , (14.30) where ( Y ( a ) 0 ) − 1 = 0. See (2.11). The nonsimply laced case is similar ly written down from (2.12)-(2.15). F or g = B r , ( Y ( a ) m ) 2 = (1 + Y ( a − 1) m )(1 + Y ( a +1) m ) (1 + ( Y ( a ) m − 1 ) − 1 )(1 + ( Y ( a ) m +1 ) − 1 ) (1 ≤ a ≤ r − 2) , ( Y ( r − 1) m ) 2 = (1 + Y ( r − 2) m )(1 + Y ( r ) 2 m − 1 )(1 + Y ( r ) 2 m ) 2 (1 + Y ( r ) 2 m +1 ) (1 + ( Y ( r − 1) m − 1 ) − 1 )(1 + ( Y ( r − 1) m +1 ) − 1 ) , ( Y ( r ) 2 m ) 2 = 1 + Y ( r − 1) m (1 + ( Y ( r ) 2 m − 1 ) − 1 )(1 + ( Y ( r ) 2 m +1 ) − 1 ) , ( Y ( r ) 2 m +1 ) 2 = 1 (1 + ( Y ( r ) 2 m ) − 1 )(1 + ( Y ( r ) 2 m +2 ) − 1 ) . (14.31) F or g = C r , ( Y ( a ) m ) 2 = (1 + Y ( a − 1) m )(1 + Y ( a +1) m ) (1 + ( Y ( a ) m − 1 ) − 1 )(1 + ( Y ( a ) m +1 ) − 1 ) (1 ≤ a ≤ r − 2) , ( Y ( r − 1) 2 m ) 2 = (1 + Y ( r − 2) 2 m )(1 + Y ( r ) m ) (1 + ( Y ( r − 1) 2 m − 1 ) − 1 )(1 + ( Y ( r − 1) 2 m +1 ) − 1 ) , ( Y ( r − 1) 2 m +1 ) 2 = 1 + Y ( r − 2) 2 m +1 (1 + ( Y ( r − 1) 2 m ) − 1 )(1 + ( Y ( r − 1) 2 m +2 ) − 1 ) , ( Y ( r ) m ) 2 = (1 + Y ( r − 1) 2 m − 1 )(1 + Y ( r − 1) 2 m ) 2 (1 + Y ( r − 1) 2 m +1 ) (1 + ( Y ( r ) m − 1 ) − 1 )(1 + ( Y ( r ) m +1 ) − 1 ) . (14.32) 118 F or g = F 4 , ( Y (1) m ) 2 = 1 + Y (2) m (1 + ( Y (1) m − 1 ) − 1 )(1 + ( Y (1) m +1 ) − 1 ) , ( Y (2) m ) 2 = (1 + Y (1) m )(1 + Y (3) 2 m − 1 )(1 + Y (3) 2 m ) 2 (1 + Y (3) 2 m +1 ) (1 + ( Y (2) m − 1 ) − 1 )(1 + ( Y (2) m +1 ) − 1 ) , ( Y (3) 2 m ) 2 = (1 + Y (2) m )(1 + Y (4) 2 m ) (1 + ( Y (3) 2 m − 1 ) − 1 )(1 + ( Y (3) 2 m +1 ) − 1 ) , ( Y (3) 2 m +1 ) 2 = 1 + Y (4) 2 m +1 (1 + ( Y (3) 2 m ) − 1 )(1 + ( Y (3) 2 m +2 ) − 1 ) , ( Y (4) m ) 2 = 1 + Y (3) m (1 + ( Y (4) m − 1 ) − 1 )(1 + ( Y (4) m +1 ) − 1 ) . (14.33) F or g = G 2 , ( Y (1) m ) 2 = (1 + Y (2) 3 m − 2 )(1 + Y (2) 3 m − 1 ) 2 (1 + Y (2) 3 m ) 3 (1 + Y (2) 3 m +1 ) 2 (1 + Y (2) 3 m +2 ) (1 + ( Y (1) m − 1 ) − 1 )(1 + ( Y (1) m +1 ) − 1 ) , ( Y (2) 3 m ) 2 = 1 + Y (1) m (1 + ( Y (2) 3 m − 1 ) − 1 )(1 + ( Y (2) 3 m +1 ) − 1 ) , ( Y (2) 3 m +1 ) 2 = 1 (1 + ( Y (2) 3 m ) − 1 )(1 + ( Y (2) 3 m +2 ) − 1 ) , ( Y (2) 3 m +2 ) 2 = 1 (1 + ( Y (2) 3 m +1 ) − 1 )(1 + ( Y (2) 3 m +3 ) − 1 ) . (14.34) The level ℓ restr icted consta nt Y-system for g is obtained from (14 .30)-(14.34) by setting ( Y ( a ) t a ℓ ) − 1 = 0 and naturally restricting the v ariables { Y ( a ) m | ( a, m ) ∈ H } to { Y ( a ) m | ( a, m ) ∈ H ℓ } . ( H ℓ is deﬁned in (14.2).) F or the TBA ana lysis, it is useful to recognize that the level ℓ restricted co nstant Y-system is expr essed in terms o f the 0th F ourier comp onent ( x = 0) o f the TBA kernels. W e prepare the notations for them. ¯ C a mn = 2 ˆ K mn a (0) , ( ¯ C a mn ) 1 ≤ m,n ≤ ℓ a − 1 = Car tan matr ix of A ℓ a − 1 , (14.35) K mk ab = ˆ K mk ab (0) =  min( t b m, t a k ) − mk ℓ  ( α a | α b ) , (14.36) P mk ab = ˆ P mk ab (0) = t ab t a δ t b m,t a k + t b − t a X j =1 j  δ t b t a ( m +1) − j, k + δ t b t a ( m − 1)+ j, k  , (14.37) J km ba = ˆ J km ba (0) = 1 2 ℓ b − 1 X n =1 ¯ C b kn K mn ab = δ ab δ mk − 1 2 N ab P km ba = − 1 2 G am,bk , (14.38) where (14.14) – (1 4.19) are used. G am,bk is deﬁned in (13.51). The sum P t b − t a j =1 in (14.37) is to be unders to o d as zer o if t a ≥ t b as in (14.16). Note that K mk ab = K km ba 119 but P mk ab 6 = P km ba and J mk ab 6 = J km ba in general. W e hav e P mk ab ∈ Z . F rom (14.35), the sp ecialization x = 0 of (14.17) gives ℓ a − 1 X n =1 ˆ A mn aa (0) ¯ C a nk = δ mk . (14.39) Using N ab and P mk ab in the a b ove, the level ℓ restricted cons tant Y-s y stem is expressed uniformly for all g as ( Y ( a ) m ) 2 = Q ( b,k ) ∈ H ℓ (1 + Y ( b ) k ) N ab P mk ab (1 + ( Y ( a ) m − 1 ) − 1 )(1 + ( Y ( a ) m +1 ) − 1 ) (( a, m ) ∈ H ℓ ) , (14.40 ) where ( Y ( a ) 0 ) − 1 = 0. This is easily s een from (14.29). T he unrestric ted version is similarly presented by replacing H ℓ here with H . The level ℓ res tricted constant Y-system is expr essed in several guises: ℓ a − 1 X n =1 ˆ K mn a (0) ln  1 + ( Y ( a ) n ) − 1  = X ( b,k ) ∈ H ℓ ˆ J mk ab (0) ln  1 + Y ( b ) k  , (14.4 1) f ( a ) m = Y ( b,k ) ∈ H ℓ (1 − f ( b ) k ) K mk ab , where f ( a ) m = Y ( a ) m 1 + Y ( a ) m . (14.4 2) The for m (14.4 1) directly follows from (14.28) and shows up na turally as the TBA equation in a certa in as y mptotic limit. See (15 .18). On the other ha nd, (14 .4 2) is deduced fro m (14.3 5) and (14.38). It is rela ted to the conjectura l q -ser ies formula [106] for the s tring function c ℓ Λ 0 λ ( q ) [1 1] of the level ℓ v acuum mo dule o f ˆ g up to a power of q : ∞ Y j =1 (1 − q j ) − rank g X { N ( a ) m } q 1 2 P ( a,m ) , ( b,k ) ∈ H ℓ K mk ab N ( a ) m N ( b ) k Q ( a,m ) ∈ H ℓ (1 − q )(1 − q 2 ) · · · (1 − q N ( a ) m ) . (1 4.43) The outer sum is ov er N ( a ) m ∈ Z ≥ 0 such that P ( a,m ) ∈ H ℓ mN ( a ) m α a ≡ λ mo d ℓ P a ∈ I Z t a α a . In fact, the crude approximation of the extremum condition on the s umma nd is q P ( b,k ) K mk ab N ( b ) k = 1 − q N ( a ) m , which is cast into (14 .42) upon s etting q N ( a ) m = 1 − f ( a ) m . The level ℓ restric ted co ns tant Y-system is the set of | H ℓ | algebra ic equatio ns o n the same num b er of unknowns { Y ( a ) m | ( a, m ) ∈ H ℓ } . With rega r d to its s o lution, the uniqueness of the p ositive real one (Theor em 5.1) is fundamental. The concrete construction of the solution is a sub ject of the subseq uent sections 14.5 and 14 .6. 14.5. Relation with Q- s ystem. Reca ll that the unr estricted Q -system for g (13.4 5) is ( Q ( a ) m ) 2 = Q ( a ) m − 1 Q ( a ) m +1 + ( Q ( a ) m ) 2 Y ( b,k ) ∈ H ( Q ( b ) k ) − 2 J ba km , (14 .44) where we hav e replaced the notation of the p ow er G am,bk by (14.38). Giv en ℓ ∈ Z ≥ 1 , we deﬁne the level ℓ restricted Q- system for g to b e the r elations o btained from 120 (14.44) by restricting the v ariables Q ( a ) m to those with ( a, m ) ∈ H ℓ by imp os ing Q ( a ) ℓ a = 1. Thus it reads ( Q ( a ) m ) 2 = Q ( a ) m − 1 Q ( a ) m +1 + ( Q ( a ) m ) 2 Y ( b,k ) ∈ H ℓ ( Q ( b ) k ) − 2 J ba km for ( a, m ) ∈ H ℓ . (14.4 5) Prop ositio n 14. 1. Su pp ose Q ( a ) m satisﬁes the level ℓ r est ricte d Q-syst em for g . Then Y ( a ) m = ( Q ( a ) m ) 2 Q ( b,k ) ∈ H ℓ ( Q ( b ) k ) − 2 J km ba Q ( a ) m − 1 Q ( a ) m +1 (14.46) is a solution of the level ℓ r estricte d c onst ant Y- system for g . The same holds b etwe en the unr estricte d Q- system and t he unr estricte d c onstant Y-system if the pr o duct Q ( b,k ) ∈ H ℓ in (14.46) is r eplac e d by Q b ∈ I ,k ≥ 1 . This is a corollar y (cons tant version) of Theorem 2.5. F o r instance in the re - stricted case, it can also b e veriﬁed direc tly by noting 1 + ( Y ( a ) m ) − 1 = Y ( b,n ) ∈ H ℓ ( Q ( b ) k ) 2 J nm ba , 1 + Y ( b ) k = ℓ b − 1 Y n =1 ( Q ( b ) n ) ¯ C b kn , (14.47) where ¯ C b kn is deﬁned by (14 .35). By virtue of (14.42), the ass ertion is re duce d to 2 J nm ba = P ℓ b − 1 n =1 ¯ C b kn K mk ab , which indeed holds by (14.38). F or g simply la ced, (14.46) reads Y ( a ) m = Q b ∈ I : C ab = − 1 Q ( b ) m Q ( a ) m − 1 Q ( a ) m +1 . (14.48 ) 14.6. Q ( a ) m at ro ot o f unity. W e ﬁx the level ℓ ∈ Z ≥ 1 . Let χ ( V ω ) b e the character of the irr educible ﬁnite dimensional r epresentation V ω of g with hig hest weigh t ω ∈ P a ∈ I Z ≥ 0 ω a . W e intro duce the following sp ecia lization of χ ( V ω ): dim q V ω = Y α ∈ ∆ + sin π ( α | ω + ρ ) ℓ + h ∨ sin π ( α | ρ ) ℓ + h ∨ , (14.49) where h ∨ is the dual Co xeter nu m be r (2.3), ∆ + is the set o f positive roo ts o f g and ρ = 1 2 P α ∈ ∆ + α = P a ∈ I ω a . The quantit y Q α ∈ ∆ + [( α | ω + ρ )] q t [( α | ρ )] q t is a q -a nalog of the dimension of V ω . Th us (14 .49) is the q -dimension at the ro o t of unity q = exp( π √ − 1 t ( ℓ + h ∨ ) ). By Prop os itio n 13 .1 0, w e k now that the cla ssical character of the Kir illov- Reshetikhin module Q ( a ) m = res χ q ( W ( a ) m ) satisﬁes the unrestricted Q-system. As shown in (13.62) a nd (13.59), res χ q ( W ( a ) m ) is a linea r co m bination of v arious χ ( V ω )’s. The sp ecializa tion of res χ q ( W ( a ) m ) to the q -dimension will b e denoted b y dim q res W ( a ) m . By the deﬁnition, Q ( a ) m = dim q res W ( a ) m still satisﬁes the unrestricted Q- system. F urthermor e, it se e ms to match the level tr unca tion as follows. 121 Conjecture 14.2 . Q ( a ) m = dim q res W ( a ) m satisﬁes t he level ℓ r est ricte d Q-system. Mor e st r ongly, t he fol lowing pr op erties hold for any a ∈ I : Q ( a ) m = Q ( a ) ℓ a − m for 0 ≤ m ≤ ℓ a , (14.50) Q ( a ) m < Q ( a ) m +1 for 0 ≤ m < [ ℓ a / 2] , (14.51) Q ( a ) ℓ a + j = 0 for 1 ≤ j ≤ t a h ∨ − 1 , (14.52) wher e [ ℓ a / 2] is the lar gest inte ger n ot exc e e ding ℓ a / 2 (not q -inte ger). Remark 14.3. Co njecture 14 .2 implies Q ( a ) m > 0 for all ( a, m ) ∈ H ℓ . Thus Y ( a ) m constructed by (14.46) with the substitution Q ( a ) m = dim q res W ( a ) m is real p os itive for all ( a, m ) ∈ H ℓ . Therefo re it must coincide with the unique solution characterized in Theorem 5.1. W e note that (14.50) implies Q ( a ) ℓ a = Q ( a ) 0 = 1 ; therefore, j = 1 cas e of (14.52) as well b ecaus e of the Q-sys tem relation ( Q ( a ) ℓ a ) 2 = Q ( a ) ℓ a − 1 Q ( a ) ℓ a +1 + Q b ( 6 = a ) ( Q ( b ) ℓ b ) − C ab and the fact that Q ( a ) ℓ a − 1 6 = 0 by (14 .51). Example 1 4.4. F or g = A r , one has Q ( a ) m = dim q res W ( a ) m = dim q V mω a from (13.63). Thus Q ( a ) m = a Y i =1 r +1 − a Y j =1 sin π ( m + i + j − 1) ℓ + r +1 sin π ( i + j − 1) ℓ + r +1 . (14.53) The pr op erty (14.50) and Q ( a ) m > 0 for ( a, m ) ∈ H ℓ are ea sily chec ked. Subs titution of this into (14 .48) gives the rea l p ositive so lution of the level ℓ restricted co nstant Y-system: Y ( a ) m = sin π a ℓ + r +1 sin π ( r +1 − a ) ℓ + r +1 sin π m ℓ + r +1 sin π ( ℓ − m ) ℓ + r +1 , 1 + Y ( a ) m = sin π ( a + m ) ℓ + r +1 sin π ( a + ℓ − m ) ℓ + r +1 sin π m ℓ + r +1 sin π ( ℓ − m ) ℓ + r +1 . (14.54 ) Obviously ( Y ( a ) 0 ) − 1 = ( Y ( a ) ℓ ) − 1 = 0 and Y ( a ) m > 0 hold for ( a, m ) ∈ H ℓ . When r = 1, this reduce s to Y (1) m in Example 5.3. One of the mos t remark able features of the level ℓ restric ted cons ta nt Y-system and Q-system is their connection with the dilog arithm ident it y (5.5) in Theorem 5 .2. The LHS emerg es fr o m the TBA analysis (Section 15). The Y ( a ) m in the dilog a rithm is characterized by the Y-system as in Theorem 5 .1 or co nstructed by the Q-s y stem as in Remark 14.3. 14.7. Bibli ographical notes. The idea of conv er ting TBA eq ua tions into diﬀer- ence equations (Y-system) as describ ed in this se c tio n w a s put in to practice b y [3] for fac torized scattering theorie s descr ibing int egrable p erturbations of co nformal ﬁeld theories. The TBA equation treated there co rresp onds to the simply laced g with level ℓ = 2 in the terminology here up to the driving ter m. Ther e a r e n umer ous Y-systems or r e lated nonlinear integral eq uations in the simila r TBA appro aches to v arious integrable ﬁeld theories, e.g. [5, 2 6 5, 266, 26 7, 2 6 8, 2 69, 270]. The Y- systems co nsidered her e app ear as typical building blo cks in these theor ies in many cases. 122 There a re a ls o exotic v ariants a nd applicatio ns of Y-systems r e la ted to T ak aha shi- Suzuki’s contin ued fr action TBA [271] in the context of p olymers [272], the sine- Gordon mo del [273] a nd the T-system for XXZ mo del [27 4]. Intricate examples of T and Y-systems are also work ed out for the dilute A L mo dels [275]. With rega rd to the Q-system, there are conjectures concer ning more general sp ecialization than dim q and re lated dilog arithm s um r ule s . See [1, app endix A], [134, app e ndix D], [4] and [101, section 1.4]. 15. TBA anal ysis of RSOS mod els W e diges t the TBA analysis of the U q ( ˆ g ) B ethe equation, which is a na tur al candidate for the level ℓ critical restr icted solid-on- s olid (RSOS) mo de l ass o ciated with the representation W ( p ) s of U q ( ˆ g ) ( ℓ ∈ Z ≥ 1 , ( p, s ) ∈ H ℓ (14.2)). The basic features of the model hav e been sketc hed in Sectio n 3 .3. The deriv ation of high temper ature entrop y a nd central charges in tw o critical regimes is outlined. The level ℓ restr ic ted Q-sy s tem, the co nstant Y-system and the dilogarithm identit y describ ed in Sections 5.1 and 14.4 – 14 .6 play a fundamen ta l role. W e ma ke a uniform treatment for genera l g elucidating the orig in o f the Y- system. The results cover rational vertex mo dels for mally as the limit ℓ → ∞ . The TBA equation (1 5.13) also applies to a n um ber of situations in other contexts, mo st notably , integrable p er tur bations of c o nformal ﬁeld theories (cf. Section 14.7) with a suitable modiﬁca tion of the LHS. Apart fro m the relatively w ell known results in the ADE ca se, a curious asp ect in nonsimply lac ed g is that the central charges in o ne o f the regimes corresp o nd to the Goddar d-Kent-Oliv e co nstruction of Virasoro modules [27 6] inv o lving the embeddings B (1) r ֒ → D (1) r +1 , C (1) r ֒ → A (1) 2 r − 1 , F (1) 4 ֒ → E (1) 6 , G (1) 2 ֒ → B (1) 3 . See (15.28)–(15.34). These res ults ha ve stimu lated notable developmen ts in crystal basis theor y of q uantum groups [262]. The conten t o f this section is based on [5 9] for ADE case and [18] for genera l g . 15.1. TBA equation. W e k eep the notations t, t a , α a , C in (2.1)–(2.2) and L, ℓ a , H ℓ in (14.1)–(14.2). The Bethe equation is the following for the unknowns { u ( a ) j | a ∈ I , 1 ≤ j ≤ n a } : sinh π 2 L  u ( a ) j − √ − 1 s t p δ ap  sinh π 2 L  u ( a ) j + √ − 1 s t p δ ap  ! N = Ω a r Y b =1 n b Y k =1 sinh π 2 L  u ( a ) j − u ( b ) k − √ − 1( α a | α b )  sinh π 2 L  u ( a ) j − u ( b ) k + √ − 1( α a | α b )  . (15.1) Here n a = N s ( C − 1 ) ap as in (3.51) with ( r i , s i ) = ( p, s ) for all i , and Ω a is a ro o t of unit y without which (1 5 .1) is ess entially the same as the Bethe equation for the vertex mo del (8.2 5) a t q = exp( π √ − 1 tL ) 38 . The Bethe equation (15 .1) is indeed v alid [59] for U q ( A (1) r ) RSOS mode l [43]. It is a well known mystery that the TBA analy s is yields supp ose dly cor rect results in the end despite that it inv o lves ar g ument s that can hardly b e justiﬁed mathematically 39 . Our ar guments in the sequel are no exception. 38 Ω a = e − α a ( H ) in the notation of (iii) in Section 8.3. 39 A more reliable deri v ation based on T- system is given in Section 16.3. 123 W e employ a string hypothesis. Supp os e that { u ( a ) j | a ∈ I , 1 ≤ j ≤ n a } is approximately gro up e d a s the union of { u ( a ) m,i + √ − 1 t − 1 a ( m + 1 − 2 n ) | 1 ≤ n ≤ m, 1 ≤ i ≤ N ( a ) m , u ( a ) m,i ∈ R } and the rest. Here u ( a ) m,i is the c e n ter of a color a length m string and N ( a ) m is the n um ber of suc h strings. Then the hypothesis is that lim N →∞ P ℓ a m =1 mN ( a ) m /n a = 1 for all a ∈ I . It means that for co lor a , only those s tr ings with length ≤ ℓ a contribute to the thermo dynamic quantities. This is a p eculiar feature in the RSOS mo del and one of the most signiﬁca nt eﬀects of the phase factor Ω a . Substituting the string forms into (15.1) and tak ing pr o duct ov er the in terna l co ordinate of strings, one gets N δ ap Θ m a  u ( a ) m,i , s t a  = I ( a ) m,i + X b ∈ I 1 ≤ k ≤ ℓ b N ( b ) k X j =1 Θ mk ab ( u ( a ) m,i − u ( b ) k,j , ( α a | α b )) . (15.2 ) Here I ( a ) m,i ∈ Z + constant, and Θ m a , Θ mk ab are deﬁned b y Θ m a ( u, ∆) = 1 2 π √ − 1 m X n =1 ln sinh π 2 L ( u + √ − 1 t − 1 a ( m + 1 − 2 n ) − √ − 1∆) sinh π 2 L ( u + √ − 1 t − 1 a ( m + 1 − 2 n ) + √ − 1∆) , (15.3) Θ mk ab ( u, ∆) = Θ km ba ( u, ∆) = k X j =1 Θ m a ( u + √ − 1 t − 1 b ( k + 1 − 2 j ) , ∆) . (15.4 ) One a ssumes tha t each solution satisfying u ( a ) m, 1 < u ( a ) m, 2 < · · · < u ( a ) m,N ( a ) m corres p o nds to an a r ray such that I ( a ) m, 1 < I ( a ) m, 2 < · · · < I ( a ) m,N ( a ) m , and in tro duces the string density ρ ( a ) m ( u ) and the hole density σ ( a ) m ( u ) for u ∼ u ( a ) m,i with lar g e eno ugh N by ρ ( a ) m ( u ) = 1 N ( u ( a ) m,i − u ( a ) m,i − 1 ) , σ ( a ) m ( u ) = I ( a ) m,i − I ( a ) m,i − 1 − 1 N ( u ( a ) m,i − u ( a ) m,i − 1 ) . (15.5) Then (15.2) is conv er ted into an in tegr al equatio n. A little insp ection of it shows a characteristic prop erty σ ( a ) ℓ a ( u ) = 0, which enables o ne to e limina te the densit y of the “longest strings” ρ ( a ) ℓ a ( u ). F or s uch calculations, it is conv enient to work in the F ourier comp onents. W e a ttach ˆ to them. See (14.9). W e sha ll ﬂex ibly pr esent formulas either in the F ourie r o r origina l v ar ia bles. By means of the basic formulas (14.20) and (14.21), the resulting in teg ral equation is express ed in the F ourier space as 40 δ pa ˆ A sm pa ( x ) = ˆ σ ( a ) m ( x ) + X ( b,k ) ∈ H ℓ ˆ K mk ab ( x ) ˆ ρ ( b ) k ( x ) for ( a, m ) ∈ H ℓ . (15.6) The “TBA kernels” A mk ab ( x ), K mk ab ( x ), etc and their useful prop erties a r e summa- rized in Section 14.2. By (14.1 7) a nd (14.15), (15.6) is also written as δ ap δ sm 2 co sh( x t a ) = ℓ a − 1 X n =1 ˆ K mn a ( x ) ˆ σ ( a ) n ( x ) + X ( b,k ) ∈ H ℓ ˆ J mk ab ( x ) ˆ ρ ( b ) k ( x ) (15.7) 40 The replacemen t ℓ → L in (14.20) and (14.21) has b ecome unnecessary here due to the elimination of ρ ( a ) ℓ a ( u ). 124 for ( a, m ) ∈ H ℓ . The equation (15 .6) or e q uiv alent ly (15.7) is the Be the equation for the string and hole densities. W e will actually consider the thermo dyna mics of the “quantum spin” chain a s- so ciated with the row to r ow transfer ma tr ix T ( p ) s ( u ) o f the RSOS mo del. W e chose its Hamiltonian density H as H = − ǫγ N ∂ ∂ u ln T ( p ) s ( u ) | u = u 0 ( ǫ = ± 1) , (15.8 ) where γ > 0 is a norma lization constant a nd ǫ = ± 1 sp eciﬁes the tw o criti- cal regimes in the RSOS mo del. The p oint u 0 is such that T ( p ) s ( u 0 ) b ecomes a cyclic shift (genera tor o f momentu m) up to an ov era ll mu ltiple, i.e . (3 .50) be- comes (scala r) Q N i =1 δ λ i ,µ i − 1 δ α i ,β i − 1 . In view of Sectio n 8.3, it is natura l to as- sume that the spectrum E of H is obtained from the deriv a tive of the top term Q p ( u − s t p ) /Q p ( u + s t p ) therein up to a n overall fac tor indep endent of the Bethe ro ots. Thus up to a n additiv e constant we get 41 E = ǫγ N ℓ p X m =1 N ( p ) m X i =1 ∂ ∂ u Θ m p  u, s t p     u = u ( p ) m,i ≃ ǫγ ℓ p X m =1 Z ∞ −∞ du ∂ ∂ u Θ m p  u, s t p  ρ ( p ) m ( u ) = ǫγ 2 π ℓ p − 1 X m =1 ˆ A sm pp ˆ ρ ( p ) m + ǫ E 0 , (15.9) where in the last step ρ ( p ) ℓ p ( u ) is eliminated a s was done for (15.6). E 0 is a co nstant whose concrete fo rm ([18, (2 .20)]) is ir relev ant in wha t follows. O n the o ther hand, the eigenv alues of the momentu m densit y P is dir ectly r elated to the top ter m its e lf, and is given as P = 2 π N ℓ p X m =1 N ( p ) m X i =1 Θ m p  u ( p ) m,i , s t p  ≃ 2 π ℓ p X m =1 Z ∞ −∞ du Θ m p  u, s t p  ρ ( p ) m ( u ) . (15.10) The Y ang-Y ang t yp e en tr opy dens it y S [6] resp onsible for the arrangement of strings and holes is S = X ( a,m ) ∈ H ℓ Z ∞ −∞ du  ( ρ ( a ) m ( u ) + σ ( a ) m ( u )) ln ( ρ ( a ) m ( u ) + σ ( a ) m ( u )) − ρ ( a ) m ( u ) ln ρ ( a ) m ( u ) − σ ( a ) m ( u ) ln σ ( a ) m ( u )  . (15.11) The ther mal equilibr ium condition at temp erature T = β − 1 is obtained b y de- manding that the fr ee ener g y density F = E − T S b e the ex tr emum with r e- sp ect to ρ ( a ) m ( u ), namely δ F / δ ρ ( a ) m ( u ) = 0 , under the c o nstraint (15.6). Setting σ ( a ) m ( u ) /ρ ( a ) m ( u ) = exp( β ǫ ( a ) m ( u )), the result reads (( a, m ) ∈ H ℓ ) ǫβ γ δ pa δ sm 4 t − 1 p cosh( t p π u/ 2) = ℓ a − 1 X n =1 Z ∞ −∞ dv K mn a ( u − v ) ln  1 + exp( β ǫ ( a ) n ( v ))  − X ( b,k ) ∈ H ℓ Z ∞ −∞ dv J mk ab ( u − v ) ln  1 + exp( − β ǫ ( b ) k ( v ))  . (15.12) 41 The sign ( − 1) in (15.8) i s absen t here since T ( p ) s ( u ) is related to ∂ ∂ v Θ m p ( v, s /t p )    v = √ − 1 u . 125 The nonlinea r in tegr al equation (15.12) is a n example o f the TBA equa tion, whic h serves a s the basis in studying thermo dyna mic quantities. By using (14.12) and (14.19) it can be slightly rea rrang ed as ǫβ γ δ pa δ sm 4 t − 1 p cosh( t p π u/ 2) = β ǫ ( a ) m ( u ) − Z ∞ −∞ dv ln h  1 + exp( β ǫ ( a ) m − 1 ( v ))  1 + exp( β ǫ ( a ) m +1 ( v ))  i 4 t − 1 a cosh( t a π ( u − v ) / 2) + X ( b,k ) ∈ H ℓ N ab Z ∞ −∞ dv h P mk ab ∗ ln  1 + exp( − β ǫ ( b ) k )  i ( v ) 4 t − 1 a cosh( t a π ( u − v ) / 2) . (15.13) When g is s imply laced, one has P mk ab ( u ) = δ mk δ ( u ) from (14.24) and (14.9). There- fore (15.13) simpliﬁes consider ably to ǫβ γ δ pa δ sm 4 co sh( πu/ 2 ) = β ǫ ( a ) m ( u ) − Z ∞ −∞ dv ln "  1+exp( β ǫ ( a ) m − 1 ( v ))  1+exp( β ǫ ( a ) m +1 ( v ))  Q b ∈ I  1+exp( − β ǫ ( b ) m ( v ))  N ab # 4 co sh( π ( u − v ) / 2) . (15.14) 15.2. High temp e rature en tropy. T he free energy density is expressed as F = ǫ E 0 − T ℓ p − 1 X m =1 Z ∞ −∞ du A sm pp ( u ) ln  1 + exp( − β ǫ ( p ) m ( u ))  (15.15) by means of (15.12), (14.1 7) and (14 .18). Let us ev alua te the high temp era ture limit of the en tropy density S high = − lim T →∞ F T . (15.16) When T → ∞ , the leading part of the asymptotic o f ǫ ( a ) m ( u ) is expected to b ecome independent of u . Thus w e set Y ( a ) m = exp( − β ǫ ( a ) m ( u )) to b e a constant a nd obtain from (15.15) that S high = ℓ p − 1 X m =1 ˆ A sm pp (0) ln  1 + Y ( p ) m  . (15.1 7) Here ˆ A sm pp (0) is the 0th F ourier comp onent of A sm pp ( u ) given by (14 .13). Similarly the TBA equation (15.12) tends to ℓ a − 1 X n =1 ˆ K mn a (0) ln  1 + Y ( a ) − 1 n  = X ( b,k ) ∈ H ℓ ˆ J mk ab (0) ln  1 + Y ( b ) k  . (15.18) This is the logar ithmic form of the level ℓ r estricted constant Y-system (14.41). Thu s we employ the solution Q ( a ) m = dim q res W ( a ) m explained in Remark 14 .3 con- structed from the q -dimension at a ro ot of unity (14.49). Substituting the latter formula in (14.47) into (15.17) a nd applying (14.39), we ﬁnd S high = ln Q ( p ) s . (15.19) 126 This is consistent with the dimension of the space of sta tes H ( N ) of the RSO S spin chain (3.49). Na mely , (15.19) implies lim N →∞ (dim H ( N )) 1 / N = dim q res W ( p ) s , (15.2 0) which agre e s with (3.5 4). 15.3. Cent ral c harges. The central charge c of the underlying confo r mal ﬁeld theory is extracted from the lo w tempera tur e asymptotics of the entropy as S low ≃ π cT 3 v F [277, 278], where v F is the F ermi velocity of the low lying ma s sless ex citations. In each regime ǫ = ± 1, the result is expres sed a s c = ǫ 6 π 2 X ( a,m ) ∈ H ℓ  L ( f ( a ) m ( ∞ )) − L ( f ( a ) m ( −∞ ))  , (15.21) where L ( x ) is the Rogers dilogar ithm (5.1 ). The nu mber f ( a ) m ( ∞ ) is the p ositive real s olution of ln f ( a ) m ( ∞ ) = P ( b,k ) ∈ H ℓ K mk ab ln(1 − f ( b ) k ( ∞ )) in the bo th regimes ǫ = ± 1 , where K mk ab is the 0 th F o urier comp onent of K mk ab (14.36). By Theorem 5.1, f ( a ) m ( ∞ ) equals f ( a ) m in (14.42) cons tructed fr o m the unique r eal p ositive solution o f the level ℓ restr icted constant Y-system for g . One the other hand, the nu m be r s f ( a ) m ( −∞ ) are to satisfy forma lly the same equation ln f ( a ) m ( −∞ ) = P ( b,k ) ∈ H ℓ K mk ab ln(1 − f ( b ) k ( −∞ )) but with extra condition f ( a ) m ( −∞ ) = (1 − ǫ ) / 2 for ( a, m ) ∈ H ǫ ℓ in the regime ǫ = ± 1 . Here the subset H ± ℓ of H ℓ is spec iﬁe d as H + ℓ = { ( p, m ) | 1 ≤ m ≤ ℓ p − 1 } , (15.22) H − ℓ = ( {  a, st a t p  | a ∈ I } s t p ∈ Z , H ( p, s ) ∩ H ℓ s t p 6∈ Z , (15.23) H ( p, s ) = {  a, s − s 0 t p  ,  a, s − s 0 t p + 1  | a ∈ I , t a = 1 } ∪ { ( a, s − s 0 ) , ( a, s ) , ( a, s − s 0 + t p ) | a ∈ I , t a = t p } , s ≡ s 0 mo d t p , 1 ≤ s 0 ≤ t p − 1 . Consequently , the equations gov er ning the remaining f ( a ) m ( −∞ )’s are split into the subsets cor resp onding to the complement H ℓ \ H ǫ ℓ . Their solutions are obtained b y restricted consta n t Y-system associa ted with v ario us subalg ebras of g and levels. The deta il ca n b e found in [18, section 3]. In an y case, the dilogarithm iden tity (5.5) suﬃces to ev alua te the sum (15.21). Below we list the results using the RHS of (5.5) L ( g , ℓ ) = ℓ dim g ℓ + h ∨ − ra nk g (15.24) as the building blo ck. R e gime ǫ = + 1. g = A r , c = L ( A r , ℓ ) − L ( A p − 1 , ℓ ) − L ( A r − p , ℓ ) 1 ≤ p ≤ r. 127 g = B r , c = L ( B r , ℓ ) − L ( A p − 1 , ℓ ) − L ( B r − p , ℓ ) 1 ≤ p ≤ r − 2 , = L ( B r , ℓ ) − L ( A p − 1 , ℓ ) − L ( A r − p , 2 ℓ ) p = r − 1 , r. g = C r , c = L ( C r , ℓ ) − L ( A p − 1 , 2 ℓ ) − L ( C r − p , ℓ ) 1 ≤ p ≤ r. g = D r , c = L ( D r , ℓ ) − L ( A p − 1 , ℓ ) − L ( D r − p , ℓ ) 1 ≤ p ≤ r − 2 , = L ( D r , ℓ ) − L ( A r − 1 , ℓ ) p = r − 1 , r. g = E 6 , c = L ( E 6 , ℓ ) − L ( D 5 , ℓ ) p = 1 , 6 , = L ( E 6 , ℓ ) − L ( A 1 , ℓ ) − L ( A 4 , ℓ ) p = 2 , 5 , = L ( E 6 , ℓ ) − 2 L ( A 2 , ℓ ) − L ( A 1 , ℓ ) p = 3 , = L ( E 6 , ℓ ) − L ( A 5 , ℓ ) p = 4 . g = E 7 , c = L ( E 7 , ℓ ) − L ( D 6 , ℓ ) p = 1 , = L ( E 7 , ℓ ) − L ( A 1 , ℓ ) − L ( A 5 , ℓ ) p = 2 , = L ( E 7 , ℓ ) − L ( A 1 , ℓ ) − L ( A 2 , ℓ ) − L ( A 3 , ℓ ) p = 3 , = L ( E 7 , ℓ ) − L ( A 4 , ℓ ) − L ( A 2 , ℓ ) p = 4 , = L ( E 7 , ℓ ) − L ( A 1 , ℓ ) − L ( D 5 , ℓ ) p = 5 , = L ( E 7 , ℓ ) − L ( E 6 , ℓ ) p = 6 , = L ( E 7 , ℓ ) − L ( A 6 , ℓ ) p = 7 . g = E 8 , c = L ( E 8 , ℓ ) − L ( E 7 , ℓ ) p = 1 , = L ( E 8 , ℓ ) − L ( A 1 , ℓ ) − L ( E 6 , ℓ ) p = 2 , = L ( E 8 , ℓ ) − L ( A 2 , ℓ ) − L ( D 5 , ℓ ) p = 3 , = L ( E 8 , ℓ ) − L ( A 3 , ℓ ) − L ( A 4 , ℓ ) p = 4 , = L ( E 8 , ℓ ) − L ( A 4 , ℓ ) − L ( A 2 , ℓ ) − L ( A 1 , ℓ ) p = 5 , = L ( E 8 , ℓ ) − L ( A 6 , ℓ ) − L ( A 1 , ℓ ) p = 6 , = L ( E 8 , ℓ ) − L ( D 7 , ℓ ) p = 7 , = L ( E 8 , ℓ ) − L ( A 7 , ℓ ) p = 8 . g = F 4 , c = L ( F 4 , ℓ ) − L ( C 3 , ℓ ) p = 1 , = L ( F 4 , ℓ ) − L ( A p − 1 , ℓ ) − L ( A 4 − p , 2 ℓ ) p = 2 , 3 , = L ( F 4 , ℓ ) − L ( B 3 , ℓ ) p = 4 . 128 g = G 2 , c = L ( G 2 , ℓ ) − L ( A 1 , 3 ℓ ) p = 1 , = L ( G 2 , ℓ ) − L ( A 1 , ℓ ) p = 2 . R e gime ǫ = − 1. If s t p ∈ Z , the central c har ge is given by c = L  g , s t p  + L  g , ℓ − s t p  − L ( g , ℓ ) + rank g . (15 .25) This is the v alue corresp o nding to the cos et pair ˆ g ⊕ ˆ g ⊃ ˆ g (15.26) level ℓ − s t p s t p ℓ. The s ituation s t p 6∈ Z can take place in nonsimply laced algebr as. The cen tr al charges for such cases a re given as follows. g = B r ( p = r, 1 ≤ s ≤ 2 ℓ − 1 , s ∈ 2 Z + 1), c = L  B r , s − 1 2  + L  B r , ℓ − s + 1 2  − L ( B r , ℓ ) + 2 r + 1 . (1 5 .27) This v alue cor r esp onds to the following cos et pair via the embedding B (1) r ֒ → D (1) r +1 : B (1) r ⊕ B (1) r ⊕ D (1) r +1 ⊃ B (1) r (15.28) level ℓ − s + 1 2 s − 1 2 1 ℓ. g = C r (1 ≤ p ≤ r − 1 , 1 ≤ s ≤ 2 ℓ − 1 , s ∈ 2 Z + 1), c = L  C r , s − 1 2  + L  C r , ℓ − s + 1 2  − L ( C r , ℓ ) + 3 r − 1 . (15.29) This v alue co rresp onds to the following coset pair v ia the embedding C (1) r ֒ → A (1) 2 r − 1 : C (1) r ⊕ C (1) r ⊕ A (1) 2 r − 1 ⊃ C (1) r (15.30) level ℓ − s + 1 2 s − 1 2 1 ℓ. g = F 4 ( p = 3 , 4 , 1 ≤ s ≤ 2 ℓ − 1 , s ∈ 2 Z + 1), c = L  F 4 , s − 1 2  + L  F 4 , ℓ − s + 1 2  − L ( F 4 , ℓ ) + 10 . (15.3 1) This v alue corres po nds to the following cos et pair via the embedding F (1) 4 ֒ → E (1) 6 : F (1) 4 ⊕ F (1) 4 ⊕ E (1) 6 ⊃ F (1) 4 (15.32) level ℓ − s + 1 2 s − 1 2 1 ℓ. g = G 2 ( p = 2 , 1 ≤ s ≤ 3 ℓ − 1 , s ≡ s 0 mo d 3 , s 0 = 1 , 2), c = L  G 2 , s − s 0 3  + L  G 2 , ℓ − s − s 0 3 − 1  + L ( A 1 , 2) − L ( G 2 , ℓ ) + 5 . (15.3 3) 129 This v alue corres po nds to the following coset pair via the embedding G (1) 2 ֒ → B (1) 3 : G (1) 2 ⊕ G (1) 2 ⊕ B (1) 3 ⊃ G (1) 2 (15.34) level ℓ − s − s 0 3 − 1 s − s 0 3 1 ℓ. In (15.27), (1 5.29), (1 5.31), (1 5.33), the contributions 2 r + 1 , 3 r − 1 , 10 , 5 other than the dilogar ithm L are equal to | H ( p, s ) | in (15 .23). These v alues of the central charges and co s et pairs a r e consistent with the a naly- ses o f RSO S mo dels [35, 5 6, 2 79] b y Baxter’s corner transfer matrix metho d [2]. F or A r level ℓ , the central charges in regime ǫ = +1 and ǫ = − 1 ar e tra nsformed to each other via the interchange ( r − 1 , ℓ , p, s ) ↔ ( ℓ, r − 1 , s, p ), which is a manifestation of the level-rank duality [5 6, 59, 280]. So far we hav e co nsidered the N site RSOS chain with the homoge ne o us quan- tum space, na mely the o ne corresp onding to ( W ( p ) s ) ⊗ N in the dual picture of vertex mo dels. One can extend the whole ana lysis to the inhomo geneous case co rresp ond- ing to ( W ( p 1 ) s 1 ⊗ · · · ⊗ W ( p k ) s k ) ⊗ N . Then the LHS of (15.12) b ecomes non v anis hing for ( a, m ) = ( p 1 , s 1 ) , . . . , ( p k , s k ), and H ǫ ℓ in (15.22) a nd (15.23) gets replaced by ∪ k i =1 ( H ǫ ℓ for ( p i , s i )). As the res ult, a broa d list of central charges is realize d, e.g. the co set pair ( ˆ g ) ⊕ k +1 ⊃ ˆ g fo r ADE case in the regime ǫ = − 1. F or more details see [18, section 4 .2 ]. Suc h a generaliza tio n has also b een consistently incorp or ated in to the crystal basis theory of one dimensiona l conﬁguration sums [262, section 3.2]. 16. T-system in use Here we present v arious applications o f the T and Y-sys tems to s olv able lattice mo dels. 16.1. Correlation l engths of v ertex mo del s. The correla tion length ξ is the simplest quant it y to characterize or dered states. It is ev aluated from the energ y ga p, which needs a length y calculation in the Bethe a ns atz approach. As an application of the T-system for transfer matrices, w e will demo nstrate a quick deriv ation of ξ [281, 134] based on the “p erio dicity at level 0”. W e consider the v ertex mo dels asso ciated with quan tum aﬃne algebra U q ( ˆ g ). The row tra nsfer matrix T ( a ) m ( u ) is given by (3.44). W e employ the par ameterization q = e − λ/t with λ > 0, wher e t = 1 , 2 , 3 is deﬁned in (2 .1). T o simplify the arg ument , we co nsider the homog eneous case ( r i , s i , w i ) = ( p, s, 1 ) for all i , thus T ( a ) m ( u ) acts on the quantum space W ( p ) s (0) ⊗ N . W e assume that t p = 1 and the sys tem size N is ev en. P os sible vertex conﬁgur ations a nd the Bo ltzmann weights are explicitly given in (3.1) for U q ( A (1) 1 ) for instance. The vertex w eights asso ciated to U q ( ˆ g ) with g o ther than A 1 hav e also been written down explicitly in some cases [49 , 48]. Based on the c o ncrete example from the U q ( A (1) 1 ) case, we assume that there is a range of the sp ectral parameter u in which the mo del is in a nt i-ferro electr ic order in the sense that those featur e s e x plained b elow a re re alized 42 . F or a more detailed account, see [13 4, s e ction 2.1]. In the ordered r egime, the ground state and the ﬁrst excited state a re a lmo st degenerate. The relev ant e nergy g ap is th us given by the ener gy diﬀerence betw een 42 In the parameterizat ion (3.1) for U q ( A (1) 1 ) case, the range is − 1 < u < 0. W e assume the same range for general U q ( ˆ g ) leaving the precise Boltzmann weigh ts corresp onding to i t unsp eciﬁed. 130 the ground state and the 2nd excited state(s). Le t T ground and T 2nd be the c orre- sp onding eigenv a lues of the tr ansfer ma tr ix. Consequently , 1 /ξ = ln( T ground /T 2nd ). W e will show that ξ is given as ξ = − 1 ln k , (16.1) where k (0 < k < 1 ) is determined by the data U q ( g ) as K ′ ( k ) K ( k ) = λh ∨ π , where h ∨ is the dual Coxeter num b er of g (2.3) as b efore. K ( k ) ( K ′ ( k )) stands for the complete elliptic integral of the ﬁr st (second) kind with mo dulus k . Recall that the unrestricted T-sys tem for g (2.22) has the form T ( a ) m ( u − 1 t a ) T ( a ) m ( u + 1 t a ) = T ( a ) m − 1 ( u ) T ( a ) m +1 ( u ) + g ( a ) m ( u ) M ( a ) m ( u ) , where the scala r function g ( a ) m ( u ) dep ends on the normaliz a tion of vertex weights. The factor M ( a ) m ( u ) is a pro duct o f T ( b ) k ’s. W e assume m ∈ t a Z > 0 and denote the eigenv alue s of T ( a ) m ( u ) also by the same s ymbol. F o r the ground state in the a nt i- ferro electric reg ime, the seco nd term on the RHS is e x po nentially larg er than the ﬁrst. So it is a go o d approximation to drop the ﬁrst term o n the RHS. The same is true for the second excited state(s). Let L ( a ) m ( u ) be the r atio of the eigenv a lue s L ( a ) m ( u ) = ( T ( a ) m ( u )) 2nd / ( T ( a ) m ( u )) ground . Then the above ar gument implies that it satisﬁes L ( a ) m ( u − 1 t a ) L ( a ) m ( u + 1 t a ) = M ( a ) m ( u ) | ∀ T ( b ) k ( v ) → L ( b ) k ( v ) . (1 6 .2) This is r egarded as the level zero r estricted T- system. F rom (2.4)–(2.10), one ca n chec k that it closes among those L ( a ) m ( u )’s with m ∈ t a Z > 0 . Moreover it enforces the following p erio dicity . (See also (3.55).) Prop ositio n 16.1 ([17], Theor em 8.8) . Supp ose t hat L ( a ) m ( u ) satisﬁes (16.2). Then the r elation L ( a ) m ( u ) L ( ω ( a )) m ( u + h ∨ ) = 1 is valid for m ∈ t a Z > 0 . H er e ω is t he involution on the index set I such that ω ( a ) = a except for the fol lowing c ases (se e Fig 1 ) 43 : g = A r , ω ( a ) = r + 1 − a, g = D r ( r : o dd), ω ( r − 1) = r , ω ( r ) = r − 1 , g = E 6 , ω (1) = 6 , ω (2) = 5 , ω (5) = 2 , ω (6) = 1 . In p articular, L ( a ) m ( u ) = L ( a ) m ( u + 2 h ∨ ) holds. See also [134, app endix A] for so me manipulation lea ding to the ab ov e res ult. Below we o nly consider a such that ω ( a ) = a . Obviously L ( a ) m ( u ) has ano ther per io dicity in the imag inary direction L ( a ) m ( u ) = L ( a ) m ( u + 2 π i λ ) 43 F or g = D r ( r : even ), we set ω ( a ) = a for any a ∈ I . 131 bec ause the vertex weigh ts a re r ational functions of z = q tu = e − λu . W e th us conclude that L ( a ) m ( u ) is doubly pe rio dic. Introduce tw o further functions h 1 , h 2 by h 1 ( u, u 0 ) = √ k sn  iλK ( k ) π ( u − u 0 )  , h 2 ( u, u 0 ) = √ k sn  iλK ( k ) π ( u − u 0 + h ∨ )  . These are meromor phic, 2 h ∨ -p erio dic, 2 π i λ -anti-perio dic functions of u and satisfy h j ( u, u 0 ) h j ( u + h ∨ , u 0 ) = 1 ( j = 1 , 2) . W e no te also that h 1 ( u, u 0 )( h 2 ( u, u 0 )) has one simple zer o (p ole) and no p oles (zeros) in the rectangle Ω := [0 , h ∨ ) × [0 , 2 π i/λ ) for u − u 0 ∈ Ω. W e denote b y { u z } , { u p } the set o f zeros 44 and p oles of L ( a ) m ( u ) in Ω, resp ectively . The ratio deﬁned b elow is analytic and no n-zero for 0 ≤ ℜ e u < h ∨ , h ( u ) = L ( a ) m ( u ) Q u z h 1 ( u, u z ) Q u p h 2 ( u, u p ) . F urthermor e we ha ve h ( u ) h ( u + h ∨ ) = 1 . (16.3) The Liouville theorem and (16.3) claim that h ( u ) = ± 1. W e thu s obtain the representation L ( a ) m ( u ) = ± Y u z √ k sn  iλK ( k ) π ( u − u z )  Y u p √ k sn  iλK ( k ) π ( u − u p + h ∨ )  . The lower excited states a re describ ed by only tw o zeros. The ab ove expres sion is then simpliﬁed to L ( a ) m ( u ) = L ( a ) m ( u ; u 1 , u 2 ) := ± k sn  iλK ( k ) π ( u − u 1 )  sn  iλK ( k ) π ( u − u 2 )  . (16.4) The lo cations of these zer os lab el the ex citations. The energy levels are almost degenerate with slight change in the loca tions o f zeros. Th us , we observe the band structure of second excited states. The correla tion function G ( R ) must sum up all the contributions from the ba nd [282] as G ( R ) − G ( ∞ ) ≃ Z du 1 Z du 2 ρ ( u 1 , u 2 )  L ( a ) m ( u ; u 1 , u 2 )  R . By ρ ( u 1 , u 2 ) we mean so me weigh t function whose explicit form is not necessa ry for our argument. Substitut ion o f (16.4) to the ab ov e leads to G ( R ) − G ( ∞ ) ≃ const · k R , showing (16.1). 44 In the Bethe ansatz, these zeros show up as “holes”. 132 16.2. Finite size corrections. E v aluation of ﬁnite s ize corr ections to the energ y sp ectra of the Hamiltonia n or the free energ y provides informa tion on the critica l behavior s uch as central charges and sca ling dimensions [27 8, 277, 283]. Numeri- cal approa ches o ften suﬀer from the smallness of sys tem size a nd other tec hnical problems such as lo garithmic corr ections. The ev aluatio n of ﬁnite size cor rections is a non trivial problem even for int egrable mo dels . The Bethe equa tion is highly transcendental and it simpliﬁes only in the thermo dynamics limit to an in tegr al equation. F or a n arbitrar y given sys tem s iz e, it is not p ossible in general to ﬁnd the exact lo ca tions of the Bethe roots. Nevertheless, there are s uccessful results in deriving ﬁnite size cor rections based on clever manipulations of Bethe equa- tions [2 8 4, 285, 286, 2 87]. Her e we demonstrate yet ano ther metho d utilizing the T-system in place of the Bethe equation following [2 88, 7]. As a concr ete e x ample we treat a level ℓ critical RSOS mo del a sso ciated with A (1) 1 in Section 3 .3 – 3.6 ( ℓ ∈ Z ≥ 2 ). Lo cal s tates o n lattice sites ra nge ov er { 1 , 2 , . . . , ℓ + 1 } . W e consider the fusion mode l in which an y neigh bo ring pair of lo cal s tates is s -admissible (1 ≤ s ≤ ℓ − 1). See (3.34) and (3.35) for the deﬁnition o f the admissibility . The transfer matr ix T s ( u ) is deﬁned by (3.38) with m, s i and v i replaced by s, s and 0, resp ectively . W e assume the system s ize N is even and treat the range − 2 ≤ u ≤ 0 (referr ed to as the reg ime I I I/IV critica l line [34]) for simplicity . W e set q = e iλ , λ = π ℓ + 2 , in the RSOS Boltzmann weigh ts accor ding to (3.3 3). Although we are concerned with s uch an is otropic mo del, the key in our appro ach is to embed it in a family of mo dels in which the admissibility (fusion degree) conditions in the hor iz o ntal a nd vertical directions can b e diﬀerent. W e consider the level ℓ fusion RSO S mo del [35] in whic h neigh b or ing states in the hor izontal direction are s -admissible while those in the vertical direction are m -admissible. The corr esp onding transfer ma trix is denoted b y T m ( u ) a nd depicted in (3.3 8) with s i = s a nd v i = 0. The ev alua tion of the ﬁnite size corr e ction to the largest eigenv alue of T s ( u ) utilizing the restricted T-system a mong { T j ( u ) } will be the main issue in the sequel. First we need to ﬁx the normalizatio ns. Let W 1 ,s be the RSO S Boltzma nn weigh ts obtained b y the s -fold fusion in the horizontal direction (cf .(3.24)). Our normalizatio n is such that W 1 ,s  a + s − 1 a − 1 a + s a     u  = [ u + s + 1] q 1 / 2 [2] q 1 / 2 . See (3.33) fo r the symbol [ u ] q 1 / 2 . F ro m no w on w e use x = ( u + 1 ) i as the s pec tr al parameter, and T m ( u ) will also b e written as T m ( x ). W e furthermor e deﬁne the normalized transfer matrices by ˜ T 0 ( x ) = 1 a nd ˜ T m ( x ) = ( T m ( x ) 1 ≤ m ≤ s, T m ( x ) Q m − s j =1 φ ( x +( m − s +1 − 2 j ) i ) s + 1 ≤ m ≤ ℓ, where we hav e introduced φ ( x ) =  sinh λx 2 sin λ  N . 133 Thanks to these nor malizations ˜ T j ( x ) is of deg ree N min( j, s ) in [ ix + · · · ] q 1 / 2 for 1 ≤ j ≤ ℓ . One then obtains the level ℓ restricted T-system for g = A 1 ˜ T j ( x − i ) ˜ T j ( x + i ) = f j ( x ) ˜ T j − 1 ( x ) ˜ T j +1 ( x ) + g j ( x ) (1 ≤ j ≤ ℓ − 1) . (1 6 .5) Here the scalar factors are given by f j ( x ) = φ ( x ) δ js and g j ( x ) = min( j,s ) − 1 Y k =0 φ ( x + ( s + j − 2 k ) i ) φ ( x − ( s + j − 2 k ) i ) . Numerical ca lculations for small s ystem sizes suggest the following analy ticit y o f ˜ T j ( x ). Assumption 16.2. ˜ T j ( x ) (1 ≤ j ≤ ℓ ) is a nalytic and nonzero in the strip |ℑ m x | ≤ 1. W e then constr uc t Y j ( x ) (1 ≤ j ≤ ℓ − 1) b y 45 Y j ( x ) = f j ( x ) ˜ T j − 1 ( x ) ˜ T j +1 ( x ) g j ( x ) . (16.6) This leads to the Y-system Y j ( x − i ) Y j ( x + i ) = (1 + Y j − 1 ( x ))(1 + Y j +1 ( x )) (1 ≤ j ≤ ℓ − 1) , (16.7) where Y 0 ( x ) = Y ℓ ( x ) = 0. The assumption o n T j ( x ) is inherited to the analy ticit y of Y j ( x ) except for Y s ( x ): Y s ( x ) has order N zero at the origin due to f s ( x ). W e th us deﬁne the mo diﬁed Y by ˜ Y j ( x ) = Y j ( x ) (tanh π 4 x ) N δ js . (1 6.8) Then the above ass umption is r ephrased as follows. Assumption 1 6.3. ˜ Y j ( x ) (1 ≤ j ≤ ℓ − 1 ) is analytic and nonzero in the strip |ℑ m x | ≤ 1. Also, 1 + Y j ( x ) is a nalytic and nonzero in the str ip |ℑ m x | ≤ ǫ for sma ll po sitive ǫ . Y and ˜ Y satisfy ˜ Y j ( x − i ) ˜ Y j ( x + i ) = (1 + Y j − 1 ( x ))(1 + Y j +1 ( x )) , (16.9 ) where a simple identit y ta nh π 4 ( x − i ) tanh π 4 ( x + i ) = 1 is used. With the a b ov e analyticity assumption, one ca n apply the F ourier transformatio n to the logar ith- mic deriv ative of the Y-sys tem 46 . After so lving it w ith resp ect to the log arithmic deriv ative of ln Y j , the inv ers e F ourier transformation followed by an integration conv erts the Y-system into the coupled integral equation (1 ≤ j ≤ ℓ − 1): ln Y j ( x ) = δ j s ln tanh N π x 4 + Z ∞ −∞ K ( x − x ′ ) ln [(1 + Y j − 1 ( x ′ ))(1 + Y j +1 ( x ′ ))] dx ′ 2 π , (16.10) K ( x ) = π 2 co sh π x 2 . (16.11) 45 W e emplo y the i n v erse of (2.24) to make the resulting i n tegral equation suitable for numerical inv estigations. 46 The deriv ative here is not essent i al. It is done just in order to ensure the conv ergence. 134 The in teg r ation constant turns out to b e zer o due to the asymptotic v alues Y j ( ∞ ) = sin( j ϑ ) sin (( j + 2) ϑ ) sin 2 ϑ =: ι ( j, ϑ ) (16 .12) with ϑ = π ℓ +2 . Up to the driving term, (16.1 0) coincides with the thermo dynamic Bethe ansatz (TBA) equation (15.1 4) for g = A 1 although they orig ina te from com- pletely diﬀeren t contexts. The asymptotic v alue (16.12) is an example of solutions to the constant Y-system. See Example 5.3 and Example 14.4. Once Y j ( x ) is obtained fro m (1 6.10), the quantit y T s ( x ) in questio n can b e ev aluated b y using the relation T s ( x − i ) T s ( x + i ) = g s ( x )(1 + Y s ( x )) . (1 6.13) Note ˜ T s ( x ) = T s ( x ). As n umerical data tells | Y s ( x ) | ≪ 1, the bulk contribution T bulk s ( x ) is deter mined by T bulk s ( x − i ) T bulk s ( x + i ) = g s ( x ). T o separate the bulk part and ﬁnite size corr ection, let T s ( x ) = T bulk s ( x ) T ﬁnite s ( x ). Then (16 .13) yields ln T bulk s ( x ) = − N Z ∞ −∞ sinh sk cosh( ℓ + 1 − s ) k k sinh 2 k s inh( ℓ + 2) k e − ikx dk , ln T ﬁnite s ( x ) = Z ∞ −∞ K ( x − x ′ ) ln(1 + Y s ( x ′ )) dx ′ 2 π . So far, all the relatio ns ar e v alid fo r arbitra r y even N . W e now pro ceed to the ev aluation of ln T ﬁnite s ( x ) in the large N limit for x ∼ O (1). The main contribution to the int egrals in (16.10) comes from x ′ ∼ ± 2 π ln 2 N . Th us it is c o nv enient to int ro duce y ± j ( θ ) := lim N →∞ Y j  ± 2 π ( θ + ln 2 N )  . The ev enness of the original Y j as a function of x implies y + j ( θ ) = y − j ( θ ). W e then arrive at simpler expres sions for N suﬃcient ly large : ln y ǫ j ( θ ) = − δ j s e − θ + Z ∞ −∞ K θ ( θ − θ ′ ) ln [(1 + y ǫ j − 1 ( θ ′ ))(1 + y ǫ j +1 ( θ ′ ))] dθ ′ 2 π , ln T ﬁnite s  2 θ π  = 2 co sh θ N Z ∞ −∞ e − θ ′ ln(1 + y + s ( θ ′ )) dθ ′ 2 π , where K θ ( θ ) := 2 π K ( 2 π θ ) = 1 cosh θ . The ﬁrst equa tion exactly coincides with the TBA eq uation in the low tempe rature limit. Thus the dilogarithm tric k (cf. [7, section 3.3], [134, section 3.2]) is natur ally applied to ev aluate ln T ﬁnite s ( x ). The ﬁnal result of the ﬁnite size cor r ection to the lar gest eigenv a lue o f T s ( x ) is given by ln T ﬁnite s  2 θ π  ≃ cosh θ 2 π N ℓ − 1 X j =1 Z y + j ( ∞ ) y + j ( −∞ )  ln(1 + y ) y − ln y 1 + y  dy = cosh θ π N ℓ − 1 X j =1  L + ( y + j ( ∞ )) − L + ( y + j ( −∞ ))  = π co sh θ 6 N  3 s s + 2 − 6 s ( ℓ + 2)( ℓ + 2 − s )  =: π co sh θ 6 N c. (16.1 4) Here L + ( y ) is r e lated to the Rogers dilogarithm L ( y ) in (5.1) by L + ( y ) = L ( y 1 + y ) = L (1) − L ( 1 1 + y ) . 135 W e hav e also used y + j ( ∞ ) = Y j ( ∞ ) = ι ( j, π ℓ +2 ) as in (16.12) while y + j ( −∞ ) = ( ι ( j, π s +2 ) 1 ≤ j ≤ s − 1 , ι ( j − s, π ℓ +2 − s ) s ≤ j ≤ ℓ − 1 . Then the dilogar ithm identit y (5.7) is applied. The quantit y c in the la st expression in (16.14) is r egarded as the central c harge [277]. This v alue agr ees with the TBA result (15.25) obtained from the low temper ature speciﬁc heat with g = A 1 and p = 1 , t p = 1. The ab ov e argument ca n be g eneralized to calculate the ﬁnite size cor r ection in excited sta tes w ith suitable mo diﬁca tions. The ma jor diﬀerence from the g round state ca se is that Ass umption 16.3 do es no t hold any longer. Instead, we as sume the following for low lying ex cited states. Assumption 16.4. There are ﬁnitely many zero s { z ( j ) α } o f ˜ T j ( x ) in the strip |ℑ m x | ≤ 1. Letting the zeros of ˜ T j ( x ) in the strip b e { z ( j ) α } , w e modify (1 6.8) as Y j ( x ) = ˜ Y j ( x )(tanh π 4 x ) N δ js Y α tanh π 4 ( x − z ( j − 1) α ) Y α ′ tanh π 4 ( x − z ( j +1) α ′ ) , which still s atisﬁes (16.9). Then it is straightforward to derive the following equa - tion v alid for arbitrar y N ln Y j ( x ) = D j + δ j s ln tanh N π 4 x + X α ln tanh π 4 ( x − z ( j − 1) α ) + X α ′ ln tanh π 4 ( x − z ( j +1) α ′ ) + Z ∞ −∞ K ( x − x ′ ) ln[(1 + Y j − 1 ( x ′ ))(1 + Y j +1 ( x ′ ))] dx ′ 2 π . (1 6.15) The in tegr ation constant D j takes a ccount of the branch o f ln tanh and it m ust be ﬁxed case b y case . F or low ly ing excita tions in the thermo dyna mic limit, it is reasona ble to ass ume | z ( j ) α | ≫ 1. Thus we employ the parameterization z ( j ) α = ( 2 π ( θ ( j ) α, + + ln 2 N ) for z ( j ) α ≫ 1 (1 ≤ α ≤ n ( j ) + ) , − 2 π ( θ ( j ) α, − + ln 2 N ) for z ( j ) α ≪ − 1 (1 ≤ α ≤ n ( j ) − ) , where n ( j ) ± denotes the num b er of z ( j ) α near ± 2 π ln 2 N . Then (16.1 5) is reduced in the limit N → ∞ to ln y ǫ j ( θ ) = D ǫ j − δ j s e − θ + X α ln tanh 1 2 ( θ − θ ( j − 1) α,ǫ ) + X α ′ ln tanh 1 2 ( θ − θ ( j +1) α ′ ,ǫ ) + Z ∞ −∞ K θ ( θ − θ ′ ) ln[(1 + y ǫ j − 1 ( θ ′ ))(1 + y ǫ j +1 ( θ ′ ))] dθ ′ 2 π . (16.16) The constants D ± j can be in genera l diﬀerent a nd depend on n ( j ) ± , etc. The subsidiary conditio ns T j ( z ( j ) α ) = 0 must also be sa tisﬁe d. This is rephras e d as Y j ( z ( j ) α + i ) = − 1 or equiv alently ln y ǫ j ( θ ( j ) α,ǫ + π 2 i ) = (2 I ( j ) α,ǫ + 1) π i 136 in terms of the branch cut integers { I ( j ) α, ± } . Thanks to (16.1 6), this is rewritten as − Z ∞ −∞ 1 sinh( θ ( j ) α,ǫ − θ − i ǫ ′ ) ln[(1 + y ǫ j − 1 ( θ ))(1 + y ǫ j +1 ( θ ))] dθ 2 π = (2 I ( j ) α,ǫ + 1) π + iD ǫ j − δ j s e − θ ( j ) α,ǫ + i X α ′ ln tanh  θ ( j ) α,ǫ − θ ( j − 1) α ′ ,ǫ 2 + π 4 i  + i X α ′ ln tanh  θ ( j ) α,ǫ − θ ( j +1) α ′ ,ǫ 2 + π 4 i  , (16.17) where ǫ ′ > 0 is inﬁnitesimally small. The ﬁnite part of the eigenv alue is now given by ln T ﬁnite s  2 θ π  = X ǫ = ± e ǫθ N  − X α e − θ ( s ) α,ǫ + Z ∞ −∞ e − θ ln(1 + y ǫ s ( θ )) dθ 2 π  . Although the expressio ns ar e more inv olved than the ground sta te case, one can still apply the diloga rithm trick to ev aluate the a b ov e. In par ticular, (16.1 7) and the elementary relations (ln tanh x 2 ) ′ = 1 / sinh x and ln tanh( x + π i 4 )+ ln tanh( − x + π i 4 ) = π i are useful. The ﬁna l result reads ln T ﬁnite s  2 θ π  = X ǫ = ± e ǫθ 2 π N ℓ − 1 X j =1  L + ( y ǫ j ( ∞ )) − L + ( y ǫ j ( −∞ )) + 1 2 D ǫ j ln 1 + y ǫ j ( ∞ ) 1 + y ǫ j ( −∞ ) − 2 π n ( j ) ǫ iD ǫ j − 2 π 2 n ( j ) ǫ X α =1 (2 I ( j ) α,ǫ + 1)  . (1 6.18) The ab ov e de r iv ation is bas e d on the ﬁr st principle . Howev er it lacks a general prescription to determine the integration consta nt s and to cho ose the branc h cut int egers. With r egard to this, a n interesting observ ation has b een made in [7, 289]. It is poss ible to absorb the additio na l driving terms in (16 .15) to integrals by adopting deformed contours L j as ln Y j ( x ) = D j + δ j s ln tanh N π 4 x + Z L j − 1 K ( x − x ′ ) ln(1 + Y j − 1 ( x ′ )) + Z L j +1 K ( x − x ′ ) ln (1 + Y j +1 ( x ′ )) dx ′ 2 π . Then the ev alua tion of the ﬁnite s ize cor rection go es para llel to the cas e of the largest eigenstate. The diﬀerences lie in the asymptotic v alues of y ǫ j ( x ) and the non trivial homotopy in the in tegration contours o f L j . The authors of [7, 289] hav e found empirical rules for the choice of ho motopy and int egration constants to repro duce known scaling dimensio ns from conformal ﬁeld theories. W e have seen that the T-system pr ovides an eﬃcient to o l in the a nalysis of ﬁnite size cor rections. It enables one to analytica lly calcula te the cen tra l c har ge (16.14) in the gr ound state. The scaling dimensions of relev ant oper ators can also be obtained by use o f the result in excited states (16.18). The ab ov e calculatio n of the ﬁnite size correction of the largest eigenv alue has b een ge ne r alized to RSO S mo dels ass o ciated with g in [1 34, s ection 3] up to analyticity argument on auxilia ry functions. 137 16.3. Quan tum transfer matrix approac h. According to Matsubara , ﬁnite siz e correctio ns and low temp erature asymptotics ar e dual pictures of the s a me ph ysical characteristics of a tw o dimensional system o n an inﬁnite cylinder of circumference N = β . Here N is the system size in the former picture and β is the inv erse temper ature in the latter. Our a nalyses o f the U q ( A (1) 1 ) RSOS mo de l in Section 15 and Section 16 .2 have b een done alo ng these tw o p oints of view. Wh at is remark able there is that b eyond the formal co incidence of the tw o pictures, the t wo entirely diﬀerent approaches end up with es sentially the same integral equation of TBA t yp e . One then exp ects a framework to trea t the ﬁnite temp erature pro ble m in the same manner as the ﬁnite size corr ections without recourse to string hypo thesis. As w e will see in the sequel, the Quantum T ra nsfer Matrix (QTM) appro ach [290] oﬀers suc h a scheme. F or a further detail, see the recent r eviews [291, 292]. QTM utilizes the equiv ale nce b etw een d + 1 dimensiona l classical mo dels and d dimensional qua ntu m system [29 3]. T o b e concrete, we arg ue along the 1 D spin 1 / 2 XXZ mo del as a prototypical integrable lattice system. H = J 4 N X j =1  σ x j σ x j +1 + σ y j σ y j +1 + ∆( σ z j σ z j +1 + 1)  = N X j =1 ˆ h j,j +1 , (16.19) where σ a ( a = x, y , z ) ar e the Pauli matrices. The p er io dic b oundary condition im- plies σ a N +1 = σ a 1 . The anisotr opy is parameterized as ∆ = co s λ . The Hamiltonian acts on “ the physical space” V phys := N N j =1 V j where V j denotes the j th cop y of C 2 = C e + ⊕ C e − . The main sub ject here is to calculate the partition function exactly Z 1d ( β , N ) = T r V phys e − β H . It would b e nice if this task ca n b e done for an y ﬁnite N , althoug h we do no t ha ve a satis fa ctory pr ogres s at present. W e th us concentrate on the ev aluatio n of the free energy per site in the thermo dynamic limit f = − lim N →∞ 1 β N ln Z 1d ( β , N ) . W e in tro duce the s ix vertex model on the 2D square la ttice. Let R ( u, v ) b e the U q ( A (1) 1 ) R matrix (in a conv ention diﬀerent from (3.1)): R ( u, v ) =     a ( u, v ) b ( u, v ) c ( u, v ) c − 1 ( u, v ) b ( u, v ) a ( u, v )     a ( u, v ) = [2 + u − v ] q 1 / 2 [2] q 1 / 2 , b ( u, v ) = [ u − v ] q 1 / 2 [2] q 1 / 2 , c ( u, v ) = q − u − v 2 , q = e iλ . Deﬁne the matrix element R αγ β δ by R ( u, v ) = X α,β ,γ ,δ =1 , 2 R αγ β δ ( u, v ) E α,β ⊗ E γ , δ . The index 1(2) r efers to e + ( e − ) in Fig. 4. The arr ows are assigned in order to distinguish this R matrix fro m other R matr ices that will appea r below. By R j,j +1 ( u, v ) w e mean the R matrix acting non trivially o nly on the tenso r pro duct 138 Rab u u u u u u u v v v v v v v b ( u, v ) b ( u, v ) a ( u, v ) a ( u, v ) c ( u, v ) c − 1 ( u, v ) + + + + + + + + + + + + − − − − − − − − − − − − α β γ δ R αγ β δ ( u, v ) Figure 4. A graphic r epresentation fo r R αγ β δ ( u, v ). The sp ectral parameter u ( v ) is asso ciated to horizo nt al (vertical) lines. Rab u u u u u u u v v v v v v v b ( v , u ) b ( v , u ) a ( v , u ) a ( v , u ) c ( v , u ) c − 1 ( v , u ) + + + + + + + + + + + + − − − − − − − − − − − − α β γ δ e R αγ β δ ( u, v ) Figure 5. A graphic r epresentation fo r e R αγ β δ ( u, v ). The sp ectral parameter u ( v ) is asso ciated to horizo nt al (vertical) lines. V j ( u ) ⊗ V j +1 ( v ). W e introduce the r ow to row (R TR) tra nsfer matrix T R TR ( u ) ∈ End( V phys ) by T R TR ( u ) = T r a ( R a,N ( u, 0 ) R a,N − 1 ( u, 0 ) · · · R a, 1 ( u, 0 )) , (16.20 ) where the s ubs cript “ a” s tands for the auxiliary space. With the lattice tra nslation e iP shifting the sites by o ne, the Baxter-L ¨ uscher formula [5 2] T R TR ( u ) = e iP  1 + λu J sin λ H + O ( u 2 )  (16.21) holds. With a rota ted R matrix e R αγ β δ ( u, v ) = R γ β δα ( v , u ) (Fig. 5 ), we introduce a rotated transfer matrix e T R TR ( u ) ∈ End( V phys ) by e T R TR ( u ) = T r a  e R a,N ( − u, 0) e R a,N − 1 ( − u, 0) · · · e R a, 1 ( − u, 0)  . The expa ns ion analog ous to (16.2 1) holds as e T R TR ( u ) = e − iP  1 + λu J sin λ H + O ( u 2 )  . W e thus obtain an impor tant identit y Z 1d ( β , N ) = T r V phys e − β H = lim M →∞ T r V phys  T double ( u = u M ) M 2  , (16.22) 139 u u − u − u M N 0 0 0 0 Figure 6. Fictitious tw o dimensional system where T double ( u ) := T R TR ( u ) e T R TR ( u ) and u M = − β J sin λ M λ . (16.23) The RHS of (16.22) can b e in ter preted as a partition function of a 2D clas sical system deﬁned on M × N sites (Fig. 6) Z 1d ( β , N ) = lim M →∞ Z 2d classical ( M , N , u M ) . This embo dies the equiv alence b etw een d + 1 dimensional clas sical mo dels and d dimensional quantum system for d = 1. Since the s pe c tra o f T double ( u ) is gapless , we still need a trick to ev a luate Z 2d classical ( M , N , u M ). W e follow the observ ation in [290] and consider the transfer ma trix pr opaga ting in the hor izontal direction, that is, T ′ QTM ( u = u M ) which acts on a virtual spac e of size M . It was shown that this transfer matr ix pos sesses a gap b etw een the larg est (Λ 0 ) and the other eig env alues Λ j ( j ≥ 1). This is a cr ucial b e neﬁt, as one only ha s to consider the larges t eigenv a lue to ev a lua te the free energy in the thermo dynamic limit lim N →∞ Z 1 N 2d classical ( M , N , u M ) = lim N →∞  T r T ′ QTM ( u = u M ) N  1 N = lim N →∞ (Λ N 0 + Λ N 1 + · · · ) 1 N = lim N →∞ Λ 0  1 +  Λ 1 Λ 0  N + · · ·  1 N ≃ lim N →∞ Λ 0 . Although w e have made use of the integrability for simplicity in the above arg ument , the same conclusion can b e proved in a mo re general setting. Theorem 16.5 ([290]) . L et Λ 0 b e the lar gest eige nvalue of T QTM . Then the fr e e ener gy p er site is given by f = − 1 β lim M →∞ ln Λ 0 . (16.24) Two problems are s till to b e ov er come. First we must ev aluate the la rgest e igen- v alue of T ′ QTM ( u M ) in w hich interaction dep ends on the ﬁctitious system size M . Second we mu st take the “T rotter limit” M → ∞ . Both of these are highly non- trivial. Nevertheless we stress the ab ove for m ulation makes it clea r why the ﬁnite size cor rection and the ﬁnite temp era tur e problem ca n b e tr eated in the sa me wa y . T o dis ent angle the diﬃculties, we introduce a s light gener alization, a commut ing QTM T QTM ( x, u ), by ass igning the para meter ix in the “ horizontal” direction [2 9 4]. 140 Rab Rab u u u u u u u v v v v v v v b ( v , u ) b ( v , u ) a ( v , u ) a ( v , u ) c ( v , u ) c − 1 ( v , u ) + + + + + + + + + + + + − − − − − − − − − − − − α β γ δ ( R t ) αγ β δ ( u, v ) Figure 7. A gr a phic representation for ( R t ) αγ β δ ( u, v ). The s p ectr al parameter u ( v ) is asso ciated to horizo nt al (vertical) lines. W e let the tra nsp osed R matr ix R t j,k ( u, v ) [295] b e ( R t ) αγ β δ ( u, v ) = R δα γ β ( v , u ) . See Fig. 7. Then T QTM ( x, u ) is de ﬁned by T QTM ( x, u ) = T r a ( R aM ( ix, − u ) R t a,M − 1 ( ix, u ) · · · R a 2 ( ix, − u ) R t a 1 ( ix, u )) . (16 .25) The pa rameter u will alw ays b e set to u M (16.23), thus we drop its dep endence hereafter. It is the new pa rameter x that will play the role of a sp ectral parameter instead. By this we mean that tw o QTMs w ith diﬀeren t v alues of x are int ertwined by the same R matrix R a,a ′ ( ix, iy ) T a ( x ) ⊗ T a ′ ( x ′ ) = T a ( x ′ ) ⊗ T a ′ ( x ) R a,a ′ ( ix, iy ) . Here T a ( x ) denotes the mono dromy matrix asso ciated to T QTM ( x, u M ). The pro of is elemen tar y . Now w e are able to in tro duce the fusion hierarch y of comm uting transfer matric e s T j ( x ) which contains T QTM ( x, u M ) a s the ﬁrst mem b er . (The u M -dep endence will be suppr essed.) By the constructio n, they s atisfy the T-system T j ( x − i ) T j ( x + i ) = T j − 1 ( x ) T j +1 ( x ) + g j ( x ) , where g j ( x ) = T 0 ( x + ( j + 1) i ) T 0 ( x − ( j + 1) i ) with T 0 ( x ) = φ ( x + (1 + u M ) i ) φ ( x − (1 + u M ) i ) , φ ( x ) =  sinh λx 2 sin λ  M 2 . (16.26) As in Section 16.2, we need a ssumptions on the a nalyticity of T j ( x ). F or simplicity we consider the case λ → 0 for a moment. Then the n umerical analysis sugges ts Conjecture 16.6. The zer os of T j ( x ) ar e distribute d almost on the line |ℑ m x | = j + 1 . W e set Y j ( x ) = T j − 1 ( x ) T j +1 ( x ) /g j ( x ) and int ro duce its mo diﬁcation ˜ Y j ( x ) = Y j ( x ) (tanh π 4 ( x − (1 + u M ) i ) tanh π 4 ( x + (1 + u M ) i )) M 2 . (16 .27) Note that u M is a small negative quantit y . Then the conjecture is transla ted to Conjecture 16 .7. ˜ Y j ( x ) is analytic and nonzer o in the strip |ℑ m x | ≤ 1 and 1 + Y j +1 ( x ) is analytic and nonzer o in the st rip |ℑ m x | ≤ ǫ for smal l ǫ . 141 This immediately leads to the integral equation ln Y j ( x ) = δ j 1 1 2 ln h tanh M π 4 ( x − (1 + u M ) i ) tanh M π 4 ( x + (1 + u M ) i ) i + Z ∞ −∞ K ( x − x ′ ) ln [(1 + Y j − 1 ( x ′ ))(1 + Y j +1 ( x ′ ))] dx ′ 2 π , (16.2 8) where K ( x ) is deﬁned in (16.11). The M enters o nly in the ﬁrst line in (16.2 8). Therefore the T rotter limit M → ∞ can b e taken analytic al ly , g iv ing ln Y j ( x ) = δ j 1 D ( x ) + Z ∞ −∞ K ( x − x ′ ) ln[(1 + Y j − 1 ( x ′ ))(1 + Y j +1 ( x ′ ))] dx ′ 2 π ( j ≥ 1) . (16.29) where D ( x ) in the driving term is given b y D ( x ) = − β π J sin λ 2 λ co s h π 2 x . (16.30) These a re nothing but the Gaudin-T a k ahashi equatio ns for the anti-ferromagnetic Heisenberg mo del. Also , they co incide with (16.10) up to the driving ter m. The free energy per site is obtained from the so lution to the above equations as f = − 1 β Z ∞ −∞ K ( x ′ ) ln(1 + Y 1 ( x ′ )) dx ′ 2 π . Summarizing, we hav e seen that T-sys tem plays the central ro le for the quan- titative studies on b oth ﬁnite s ize sy s tem and ﬁnite temp era ture s y stem. A wider range of the parameter 0 < λ ≤ π 2 is treated in [274] under the res tr iction that the contin ued fr actional expansion of π /λ terminates at a ﬁnite s ta ge. A suitably chosen subset of the fusion Q TMs ar e shown to s atisfy a clo sed set of f unctional relations and it successfully rec overs the w ell k nown T ak ahashi-Suzuki co ntin ue d fraction TBA equation [271] without using string hypothes is. See [274] for details. 16.4. Simpl iﬁed TBA equat ions. W e contin ue o ur disc us sion on the XXZ spin chain at ﬁnite temp eratures. W e retain the deﬁnitions of the symbols such as φ ( x ) , T j ( x ) , u M , etc. in the previo us subsectio n. The TBA equa tion is a co upled set of integral equations with (ﬁnitely o r inﬁnitely) many unknown functions Y j ( x ). It is known that equations change their for ms drastically according to a sma ll change in co upling co ns tant λ [27 1]. On the other hand, we exp ect only small changes in physical qua ntities. Thus o ne may hop e a lternative formulations that are mor e stable a gainst the change in λ . Here w e present o ne such appr oach which also originates from the T-system. It is sometimes referr ed to as a simpliﬁed TBA equation [296]. The idea is co mplemen tary to the Q TM method where one pa ys atten tion to the zero s of T j ( x ). In the simpliﬁed TB A, one is concerned with s ing ularities o f a renormaliz e d T j ( x ). The latter is deﬁned by ˜ T j ( x ) = T j ( x ) φ ( x + ( j + 1 + u M ) i ) φ ( x − ( j + 1 + u M ) i ) , (16.31) 142 where φ ( x ) is deﬁned in (16.26). Note ˜ T j ( x ) p osse sses po les of or der M / 2 a t x ∼ ± ( j + 1) i . Accordingly , the ﬁr st equation of the T-system reads ˜ T 1 ( x + i ) ˜ T 1 ( x − i ) = ˜ T 2 ( x ) + b ( M ) 1 ( x ) , (16.32) b ( M ) 1 ( x ) = φ ( x + (1 − u M ) i ) φ ( x − (1 − u M ) i ) φ ( x + (1 + u M ) i ) φ ( x − (1 + u M ) i ) . (1 6.33) Let τ j ( x ) be ˜ T j ( x ) after the T ro tter limit τ j ( x ) = lim M →∞ ˜ T j ( x ) . Then τ 1 ( x ) develops singular ity at x = ± 2 i . By construction, it is per io dic under x → x + 2 p 0 i , where p 0 = π / λ . W e thus assume the expansion τ 1 ( x ) = 2 + X n ∈ Z ∞ X j =1 c j ( x − 2 i − 2 p 0 ni ) j + X n ∈ Z ∞ X j =1 ¯ c j ( x + 2 i − 2 p 0 ni ) j . (16.34) W e utilize the T-sys tem and information on the lo cations of singularities to ﬁx c j and ¯ c j . Rewrite the T r otter limit of (16.32) as τ 1 ( x + i ) = b 1 ( x ) τ 1 ( x − i ) + τ 2 ( x ) τ 1 ( x − i ) , (16.35) b 1 ( x ) = lim M →∞ b ( M ) 1 ( x ) = exp  β J sin 2 λ cosh λx − cos λ  . (16.36) The LHS p oss esses the sing ularities a t x = i, − 3 i , while only the ﬁrst term on the RHS po ssesses singularity at x = i . Cons equently we have c j = I y = i b 1 ( y ) τ 1 ( y − i ) ( y − i ) j − 1 dy 2 π i = I y =0 b 1 ( y + i ) τ 1 ( y ) y j − 1 dy 2 π i . The contour fo r the ﬁrs t integral is a small circle centered at y = i and the same circle centered at y = 0 for the seco nd. Similarly , by r ewriting (16 .3 2) in the form τ 1 ( x − i ) = b 1 ( x ) τ 1 ( x + i ) + τ 2 ( x ) τ 1 ( x + i ) , one ﬁnds ¯ c j = I y =0 b 1 ( y − i ) τ 1 ( y ) y j − 1 dy 2 π i . By substituting the ex pressions for c j , ¯ c j int o (16.34) and p erfo rming the summa tio n ov er j and n , we arr ive a t the closed in tegral equation inv o lv ing τ 1 ( x ) only: τ 1 ( x ) = 2 + λ 4 π i  I y =0 b 1 ( y + i ) coth λ 2 ( x − y − 2 i ) dy τ 1 ( y ) + I y =0 b 1 ( y − i ) coth λ 2 ( x − y + 2 i ) dy τ 1 ( y )  . Once the ab ov e equation is solved, the free ener gy is given by f = − 1 β ln τ 1 (0). It tur ned out the new equa tion works eﬃcient ly to pro duce the hig h temper a ture expansion. One ass umes τ 1 ( x ) in the form, τ 1 ( x ) = exp  ∞ X n =0 a n ( x )( β J ) n  . Then the co eﬃcient s a n ( x ) can b e iteratively determined. 143 The simpliﬁed TBA equa tio ns are a pplied in many diﬀerent c ontexts a nd they successfully provide high temp eratur e data of the mo dels [29 7, 2 9 8]. The deriv a- tion of the simpliﬁed TBA equa tio ns re q uires le ss information on the analy ticit y . Therefore it is quite eﬃcie nt when the ana lytic pro p erty is diﬃcult to investigate. The non-compa c t case is such an example. See [19 1] for the applications to certain sectors of N = 4 sup e r Y ang-Mills theory and [29 9] to thermo dyna mics of ladder comp ounds. There is how ever a price to pay . Any eig env alue of T j ( x ) satisﬁes the same equation after reno r malization. There fo re the equation itself can not select the right answer. Rather, one has to know a priori the rig ht go al to be a chieved and start from a suﬃciently near p oint to the go al in numerical approaches. The co nv erge nc e bec omes a lso problema tic in the low temp era ture r e gime and one needs to a pply , e.g. the Pad ´ e approximation to improve the accurac y . 16.5. Hybrid e quations. There is yet further approa ch to the ﬁnite size and the ﬁnite temper ature problems [295, 300, 301]. It also makes use of a ﬁnite set of un- known functions a nd diﬀerent t yp e s of in tegr al equations fro m those der ived in the previous sections. F ollowing [302], we refer to it a s NLIE (NonLinear Integral Equa- tion) 47 just in order to disting uish it from the o ther nonlinear in tegra l equations discussed hitherto. It turns out that a hybridization of TBA and NLIE is p o ssible [303]. The hybrid approa ch is esp ecially eﬃcient in dealing with thermo dy namics of higher spin XXZ mo dels as explained b elow. W e treat the integrable spin s/ 2 XXZ mo de l whose Hamiltonian H is o btained from the fusion R matrix in Section 3.1 as H = N X i =1 h i,i +1 , h i,i +1 ∝ d du P R ( k,k ) ( q u ) | u =0 , where P is the transp ositio n. A simple g e ne r alization of the arg umen t in Section 16.3 tells tha t the free energy p er s ite is obtained from the larges t v alue of QTM T s ( x = 0) c o nsisting of the R matrix acting on V s ⊗ V s . As b efore we s et q = e iλ , λ = π p 0 and assume s ≤ p 0 − 1 . As in Section 1 6 .3, we in tro duce the auxiliary Q T M T j ( x ). This time, w e prepare only a ﬁnitely many one s { T j ( x ) } ℓ j =1 , where the integer ℓ is arbitr ary as far as it is in the rang e s ≤ ℓ ≤ 2 p 0 − s − 2 . (16.37) With a suitable normaliza tion, we ha ve the T-system T j ( x + i ) T j ( x − i ) = f j ( x ) T j − 1 ( x ) T j +1 ( x ) + g j ( x ) (1 ≤ j ≤ s − 1) , (16.38 ) g j ( x ) := min( j,s ) − 1 Y m =0 Φ( x − ( s + j − 2 m ) i )Φ( x + ( s + j − 2 m ) i ) , Φ( x ) :=  [ x + (1 + u ) i ] q 1 2 [ x − (1 + u ) i ] q 1 2  M / 2 , 47 The equation ﬁrst app eared i n the con text of ﬁnite size problem i n the XXZ mo del [287]. The simplest case is sometimes referr ed to as the DDV equation in the cont ext of inte grable ﬁeld theories. 144 where f j ( x ) = Φ( x ) δ js . This lo oks formally the sa me as (16 .5), although the mean- ing o f ℓ is diﬀerent here. As us ual we set Y j ( x ) = f j ( x ) T j − 1 ( x ) T j +1 ( x ) /g j ( x ) and deﬁne its slight mo diﬁcation g eneralizing (16.27) as ˜ Y j ( x ) = Y j ( x )  tanh π 4 ( x + (1 + u ) i ) ta nh π 4 ( x − (1 + u ) i )  M 2 δ js . Then, the modiﬁed Y-system (16.9) holds for 1 ≤ j ≤ ℓ − 2. In addition we introduce the auxilia ry functions b ( x ) , ¯ b ( x ). They are deﬁned by the combination of the terms a ppe aring in the dress e d v acuum for m of T ℓ ( x ). F o r general n , the dressed v acuum form reads T n ( x ) = P n +1 m =1 λ ( n ) m ( x ), where λ ( n ) m ( x ) = Φ ( n ) m ( x ) Q ( x + ( n + 1) i ) Q ( x − ( n + 1) i ) Q ( x + (2 m − n − 1) i ) Q ( x + (2 m − n − 3) i ) , Φ ( n ) m ( x ) = Q s − 1 r =0 Φ( x + (2 m − n − s − 1 + 2 r ) i ) Q max( s − n, 0) r =1 Φ( x − ( s + 1 − n − 2 r ) i ) . Then the auxiliary functions are deﬁned by b ( x ) = λ ( ℓ ) 1 ( x + i ) + · · · + λ ( ℓ ) ℓ ( x + i ) λ ( ℓ ) ℓ +1 ( x + i ) ( − 1 ≤ ℑ m x < 0) , ¯ b ( x ) = λ ( ℓ ) 2 ( x − i ) + · · · + λ ( ℓ ) ℓ +1 ( x − i ) λ ( ℓ ) 1 ( x − i ) (0 < ℑ m x ≤ 1 ) , which a re ass umed to be analytic and nonzer o in the str ips indica ted in the paren- theses for the largest eigenv a lue of the QTM T s ( x ). W e a lso in tro duce B ( x ) = 1 + b ( x ) , ¯ B ( x ) = 1 + ¯ b ( x ) in each analytic strips. There are nice relations among them, e.g. Y ℓ − 1 ( x − i ) Y ℓ − 1 ( x + i ) = (1 + Y ℓ − 2 ( x )) B ( x ) ¯ B ( x ) , b ( x ) = Φ( x ) δ ℓs Q s r =1 Φ( x + ( ℓ − s + 2 r ) i ) Q ( x + ( ℓ + 2) i ) Q ( x − ℓi ) T ℓ − 1 ( x ) , ¯ b ( x ) = Φ( x ) δ ℓs Q s r =1 Φ( x − ( ℓ − s + 2 r ) i ) Q ( x − ( ℓ + 2) i ) Q ( x + ℓi ) T ℓ − 1 ( x ) , which can b e easily chec ked b y using the deﬁnitions. 145 By use of the analyticity a ssumptions, it is str a ightforw a r d to derive the following equations after the limit M → ∞ . ln Y j ( x ) = δ j s D ( x ) + Z ∞ −∞ K ( x − x ′ ) ln[(1 + Y j +1 ( x ′ ))(1 + Y j − 1 ( x ′ ))] dx ′ 2 π , 1 ≤ j ≤ ℓ − 2 , (16 .39) ln Y ℓ − 1 ( x ) = δ ℓ − 1 ,s D ( x ) + Z ∞ −∞ K ( x − x ′ ) ln(1 + Y ℓ − 2 ( x ′ )) dx ′ 2 π + Z C − K ( x − x ′ ) ln B ( x ′ ) dx ′ 2 π + Z C + K ( x − x ′ ) ln ¯ B ( x ′ ) dx ′ 2 π , (16.40) ln b ( x ) = δ ℓs D ( x ) + Z ∞ −∞ K ( x − x ′ ) ln (1 + Y ℓ − 1 ( x ′ )) dx ′ 2 π + Z C − F ( x − x ′ ) ln B ( x ′ ) dx ′ 2 π − Z C + F ( x − x ′ + 2 i ) ln ¯ B ( x ′ ) dx ′ 2 π x ∈ C − , (16.41) ln ¯ b ( x ) = δ ℓs D ( x ) + Z ∞ −∞ K ( x − x ′ ) ln (1 + Y ℓ − 1 ( x ′ )) dx ′ 2 π + Z C + F ( x − x ′ ) ln ¯ B ( x ′ ) dx ′ 2 π − Z C − F ( x − x ′ − 2 i ) ln B ( x ′ ) dx ′ 2 π x ∈ C + , (16.42) where C + ( C − ) is a contour just ab ov e (b elow) the real axis. The kernel K ( x ) is given in (16.11) and F is re lated to the spinon S matrix F ( x ) = Z ∞ −∞ sinh( p 0 − ℓ − 1 ) k 2 co sh k s inh k ( p 0 − ℓ ) e − ikx dk . The integration co ns tants are found to b e zero by comparing asymptotic v alues of the b o th sides and D ( x ) is deﬁned in (16.30). Obviously (16.39) is a reminiscence of the TBA type equation (16 .29), while (16.41) and (16.4 2) res emb le NLIE were it not for the ln(1 + Y ℓ − 1 ) ter m. In this sense we ca ll the a bove equatio ns hybrid. They ﬁx the v alues of Y s ( x ). The functional relations s imilar to (16.13) and the trick mentioned ar ound (16 .13) then yield the ev aluation of the free energy p er site. Remark 16.8 . The num b er ℓ is ar bitrary under the condition (16 .37). This is quite diﬀerent from “genuine” TBA equations at sp ecial λ [27 1, 274], wher e the num b er of equations is completely determined by λ . When λ → 0, we can formally put ℓ = ∞ , which rec ov ers the usual TBA eq uation in the ra tional limit as ar gued in Se c tion 16.3 for s = 1 . F or s = 1, o ne can make F ( x ) n ull by choo sing p 0 = ℓ + 1. The resulting system re pr o duces the known TBA equation corre s po nding to the level 2 restricted Y-system for D ℓ +1 for the XXZ chain. See [27 4, eq .(4.10)-eq.(4.1 2)] for example. F or ar bitrary s ∈ Z ≥ 1 , the choice ℓ = s re c overs the result in [303]. The a bove equations are n umerically stable and y ield a quick conv ergence to the unique solution. T hey are eﬃcient in the analysis of the low tempe r ature re g ime. It is a lso known that with a suitable mo diﬁca tion, o ne can der ive the equations for excited states. W e a gain hav e to pay the price. The systematic a lgorithm to construct the auxiliary functions is still lacking ex cept for g = A 1 discussed here. This remains as an int eresting future problem. 146 Ackno wledgments The author s thank Murray T. 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