Tetrahedron equations, boundary states and hidden structure of U_q(D_n^1)

TETRAHEDR ON EQUA TIONS, BOUND AR Y ST A TES AND H IDDEN STR UCTURE OF U q ( D (1) n ) SERGEY M. S ERGEEV Abstra ct. Simple p eriodic 3 d → 2 d compactification of the tetrahedron equations giv es the Y ang-Baxter equations for vari ous ev aluation represen tations of U q ( b sl n ). I n this pap er w e construct an ex ample of fixed n on-p eriod ic 3 d b ound ary conditions pro ducing a set of Y ang-Baxter equations for U q ( D (1) n ). These b oundary conditions resemble a fusion in hidden direction. The tetrahedron equation can b e view ed as a lo cal condition pro viding existence of an in- finite ser ies of Y ang-Baxter equations. In the applications to quan tum group s the metho d of tetrahedron equation is a p o werful tool for generation of R -matrices and L -op erators f or v ar- ious “higher spin” ev aluation represen tations. This has b een demonstrated in [1] for U q ( b sl n ) and in [4] for sup er-algebras U q ( b g l n | m ). The main pr inciple p ro ducing the cyclic b sl n structure is the trace in hidd en “third” direc- tion. In this pap er we introd uce another b oundary condition, a certain b oundary states still pro viding th e existence of effectiv e Y ang-Baxter equation and integrabilit y . W e shall start with a short remained of a (sup er -)tetrahedron equation and b sl n compact- ification in their elemen tary form. The simplest kno w n tetrahedron equation in the tensor pro du ct of six spaces B 1 ⊗ F 2 ⊗ · · · ⊗ F 5 ⊗ B 6 is (1) R B 1 F 2 F 3 R B 1 F 4 F 5 R F 2 F 4 B 6 R F 3 F 5 B 6 = R F 3 F 5 B 6 R F 2 F 4 B 6 R B 1 F 4 F 5 R B 1 F 2 F 3 , where F i = {| 0 i , | 1 i} i is a representa tion space of F ermi oscillator (2) f + | 0 i = | 1 i , f − | 1 i = | 0 i . Odd op erators f ± i in different co mp onents i of their tensor p r o duct anti -comm ute and ( f ± i ) 2 = 0. It is con venien t to in trod uce pro jectors (3) M i = f + i f − i , M 0 i = f − i f + i , [ f + i , f − i ] + = M 0 i + M i = 1 . Op erator M 0 i is the pro jector to v acuu m, M i is the o ccupation num b er and M 0 M = 0. Space B i stands for representat ion space of i -th cop y of q -oscillator, (4) b + b − = 1 − q 2 N , b − b + = 1 − q 2 N +2 , q N b ± = b ± q N ± 1 . 1991 Mathematics Subje ct Cl assific ation. 81Rx x,17B80. 1 2 SERG EY M. SERGEEV In this pap er w e imp ly the unitary F o c k space rep resen tation, ( b − ) † = b + , defined by (5) N | n i = | n i n , b − | 0 i = 0 , | n i = b + n p ( q 2 ; q 2 ) n | 0 i , n ≥ 0 , where ( x ; q 2 ) n = (1 − x )(1 − q 2 x ) · · · (1 − q 2 n − 2 x ). In terms of creation, an n ihilation and o ccupation n um b er op erators the R -matrices in (1) are giv en [4] by (6) R B 1 F 2 F 3 = M 0 2 M 0 3 − q N 1 +1 M 2 M 0 3 + q N 1 M 0 2 M 3 − M 2 M 3 + b − 1 f + 2 f − 3 − b + 1 f − 2 f + 3 and (7) R F 1 F 2 B 3 = M 0 1 M 0 2 + M 1 M 0 2 q N 1 +1 − M 0 1 M 2 q N 1 − M 2 M 3 + f + 1 f − 2 b − 3 − f − 1 f + 2 b + 3 . Both op erators R are unitary ro ots of unity . The constan t tetrahedron equation (1) can b e v erified in the op erator language straigh tforw ardly . Define next the “mono dromy” of R -matrices as the ord er ed pro d uct (8) R ∆ n ( B 1 F 2 ) ,F 3 = R B 1:1 F 2:1 F 3 R B 1:2 F 2:2 F 3 · · · R B 1: n F 2: n F 3 ⇌ y Y j =1 ..n R B 1: j F 2: j F 3 . Here th e con v en ient “co-pro duct” n otation stands for a tensor p o w er of corr esp onding spaces, (9) ∆ n ( B 1 ) = n ⊗ j =1 B 1: j , ∆ n ( F 2 ) = n ⊗ j =1 F 2: j . The rep eated use of (1) pro vides (10) R ∆ n ( B 1 F 2 ) ,F 3 R ∆ n ( B 1 F 4 ) ,F 5 R ∆ n ( F 2 F 4 ) ,B 6 R F 3 F 5 B 6 = R F 3 F 5 B 6 R ∆ n ( F 2 F 4 ) ,B 6 R ∆ n ( B 1 F 4 ) ,F 5 R ∆ n ( B 1 F 2 ) ,F 3 . Note the conserv ation la ws: (11) v − M 3 u − M 5  u v  N 6 R F 3 F 5 B 6 = R F 3 F 5 B 6 v − M 3 u − M 5  u v  N 6 . Multiplying (10) by the u, v -term in F 3 ⊗ F 5 ⊗ B 6 and by R − 1 F 3 F 5 B 6 , and making then the traces o ver F 3 ⊗ F 5 ⊗ B 6 , we come to th e Y ang-B axter equation (12) L ∆ n ( B 1 F 2 ) ( v ) L ∆ n ( B 1 F 4 ) ( u ) R ∆ n ( F 2 F 4 ) ( u/v ) = R ∆ n ( F 2 F 4 ) ( u/v ) L ∆ n ( B 1 F 4 ) ( u ) L ∆ n ( B 1 F 2 ) ( v ) , where (13) L ∆ n ( B 1 F 2 ) ( v ) = S tr F 3  v − M 3 R ∆ n ( B 1 F 2 ) ,F 3  , R ∆ n ( F 2 F 4 ) ( w ) = T r B 6  w N 6 R ∆ n ( F 2 F 4 ) ,B 6  . This is the case of U q ( b sl n ). Two- dimensional R -matrices (13) hav e the centers (14) J i = n X j =1 M i : j for f er m ions and J 1 = n X j =1 N 1: j for b osons. HIDDEN STRUCTURE OF U q ( D (1) n ) 3 Irreducible comp onents of R -matrices and L -op erators (13) corresp ond to fi xed v alues of J i . In particular, ∆ n ( F ) is the sum of all an tisymmetric tensor representa tions of s l n , (15) dim ∆ n ( F ) = 2 n = n X k =0 n ! k !( n − k )! . The Dirac spinor repr esentati on of D n has the same dimens ion 2 n , it is the direct su m of t w o irreducible W eyl spinors with dimensions 2 n − 1 . It is eviden t in tuitiv ely , th e structure of D n will app ear if the tot al o ccupation num b er J of ∆ n ( F ) is n ot a cen ter of L -operators and R -matrices, but all op erators preserve ju st the parit y of J . Also, since the dimension of vec tor represent ation of D n is 2 n , we need to double the num b er of b osons. Consider now t w o copies of (1) and fur ther of (10) glued in the “second” dir ection. This consideration ke eps the d esired space ∆ n ( F ) and doubles the num b er of b osons. T he rep eate d use of (1) pro vides (16) R ∆( B 1 ) F 2 ∆( F 3 ) R ∆( B 1 ) F 4 ∆( F 5 ) R F 2 F 3 B 6 R ∆ ′ ( F 3 F 5 ) B 6 = R ∆ ′ ( F 3 F 5 ) B 6 R F 2 F 3 B 6 R ∆( B 1 ) F 4 ∆( F 5 ) R ∆( B 1 ) F 2 ∆( F 3 ) , where (17) R ∆( B 1 ) F 2 ∆( F 3 ) = R B 1 F 2 F 3 R B ′ 1 F 2 F ′ 3 and R ∆ ′ ( F 3 F 5 ) B 6 = R F ′ 3 F ′ 5 B 6 R F 3 F 5 B 6 . The k ey observ ation is the existence of a family of eigen vecto rs of op erator R ∆ ′ ( F 3 F 5 ) B 6 : (18) R ∆ ′ ( F 3 F 5 ) B 6 | ψ ∆( F 3 ) ( v ) ψ ∆( F 5 ) ( u ) ψ B 6 ( u/v ) i = | ψ ∆( F 3 ) ( v ) ψ ∆( F 5 ) ( u ) ψ B 6 ( u/v ) i , where (19) ∆( F ) = F ′ ⊗ F , | ψ ∆( F ) ( v ) i = (1 + v − 1 f + ′ f + ) | 0 i , and in the unitary basis (3) (20) h 2 k + 1 | ψ B ( w ) i = 0 , h 2 k | ψ B ( w ) i = w k s ( q 4 k + 4 ; q 4 ) ∞ ( q 4 k + 2 ; q 4 ) ∞ . The normalizat ion of ψ B is giv en by (21) h ψ B ( w ) | ( b ± ) 2 m | ψ B ( w ) i = w m ( q 2+4 m w 2 ; q 4 ) ∞ ( w 2 ; q 4 ) ∞ . Considering no w a length- n chai n of (16) in the “third” direction and applying ve ctors ψ ∆( F 3 ) ( u ), ψ ∆( F 5 ) ( v ) and ψ B ( u/v ), w e come to th e Y ang-B axter equation (22) L ∆ n (∆( B 1 ) F 2 ) ( v ) L ∆ n (∆( B 1 ) F 4 ) ( u ) R ∆ n ( F 2 F 4 ) ( u/v ) = R ∆ n ( F 2 F 4 ) ( u/v ) L ∆ n (∆( B 1 ) F 4 ) ( u ) L ∆ n (∆( B 1 ) F 2 ) ( v ) without trace construction: (23) L ∆ n (∆( B 1 ) F 2 ) ( v ) = h ψ ∆( F 3 ) ( v ) | R ∆ n (∆( B 1 ) F 2 ) , ∆( F 3 ) | ψ ∆( F 3 ) ( v ) i 4 SERG EY M. SERGEEV and (24) R ∆ n ( F 2 F 4 ) ( w ) = h ψ B 6 ( w ) | R ∆ n ( F 2 F 4 ) ,B 6 | ψ B 6 ( w ) i . Matrix elemen ts of R ∆ n ( F 2 F 4 ) ( w ) can b e calculated with the help of (21) and similar identiti es. The in v arian ts of L -op erator (23) and R -matrix (24) are: the parit y of J 2 = P M 2: j , similar parit y of J 4 and (25) J 1 = n X j =1 ( N 1: j − N ′ 1: j ) . A choic e of different sp ectral parameters in bra- an d ket -v ectors in (23,24) is equiv alen t to the choic e of equal sp ectral parameters b y m eans of a gauge transformation. The structure of D n represent ation ring can b e verified explicitly by a direct calculation of matrix element s of R -matrix (24 ) for small n and c h ec k of factor p ow ers of d et( λ − R ). As to 2 n -b osons space, irr educible comp onents of ∆ n (∆( B 1 )) are in general infi nite di- mensional. Ho wev er, a c h oice of F oc k and ant i-F o c k sp ace r epresent ations, Sp ectrum( N 1: j ) = 0 , 1 , 2 , . . . and S p ectrum( N ′ 1: j ) = − 1 , − 2 , − 3 , . . . , makes ∆ n (∆( B 1 )) a direct su m of symmet- ric tensors of O (2 n ). The main resu lt of this p ap er is a step forwa rd to a classification of inte gr able b oundary c on- ditions in th ree-dimensional mo dels. A t least tw o scenarios are hitherto kn o wn: quasi-p erio dic b ound ary condition (13) and th e b oun dary states condition (23,24). Th ese conditions can b e imp osed for a la y er -to-la y er tr ansfer matrix in different directions ind ep endent ly . In b oth scenarios the sp ectral parameters of effectiv e tw o-dimensional mo dels reside the b ound ary . Also, the b ound ary admits t wists making the quant um groups classification inapplicable [3]. It worth noting one more p ossible scenario of integrable b oundary conditions: y et unknown 3 d r eflection op erators satisfying the tetrahedron r eflection equations [2]. Ac knowledgemen ts. I w ould lik e to thank all staff of the F ac ult y of Inf ormation Science for th eir supp ort. Referen ces [1] V. V. Bazhano v an d S . M. Sergeev, Zamolo dchikov’s tetr ahe dr on e quation and hidden structur e of quantum gr oups , J. Phys. A 39 (2006), no. 13, 3295–3310 [2] A. P . Isaev and P . P . Ku lish, T etr ahe dr on r efle ction e quations , Modern Phys. Lett. A 12 (1997), no. 6, 427–437 [3] S. Sergeev, An satz of Hans Bethe for a two-dimensional lattic e Bose gas , J. Phys. A 39 (2006 ), no. 12, 3035–30 45 [4] S. M. Sergeev, Sup er-tetr ahder a and sup er-algebr as , arXiv :0805.4653 , 2008. F acul ty of Informa tional Sciences and Engine ering, Uni versity of Can berra, Bruce ACT 2601 E-mail addr ess : sergey.sergeev@can berra.edu.a u