Integrable structure of melting crystal model with external potentials

In tegrable structure of melting crystal mo del with ex ternal p oten tials T oshio Nak atsu 1 and Kanehisa T ak asaki 2 ∗ 1 F acult y of Engineering, Mathematics and Ph ysics,S etsunan Univ ersit y Ik edanak amac hi, Ney aga wa, Osak a 572-850 8, Japan 2 Graduate Sc ho ol of Human and En vironmen t al Studies, Ky oto Univ ersit y Y oshida, Saky o, K y oto 606 -8501, Japan Abstract This is a review of the authors’ recen t results on an integrable structure of the melting crystal model with external potentials. The partition func- tion of this mo del is a sum ov er all plane partitions (3D Y oung diagrams). By the method of transfer matrices, this sum turn s into a sum o ver ordi- nary partitions (Y oun g dia grams), whic h may b e th ought of as a mo del of q -deformed rand om partitions. This mod el can b e further translated to the language of a complex fermion system. A fermionic realization of the qu antum torus Lie algebra is shown to underlie therein. With the aid of hidden symmetry of th is Lie algebra, the partition function of th e melting crystal model turns out to coincide, up to a simple factor, with a tau function of the 1D T oda h ierarc hy . Some related issues on 4D and 5D sup ersymmetric Y ang-Mills theories, top ological strings and the 2D T oda hierarc hy are briefly discussed. ∗ E-mail: ta k asaki@math.h.kyoto-u.ac.jp 1 Figure 1 : Melting crystal corner 1 In tro d uction The melting crystal mo del is a mo del of sta tis tica l mechanics that descr ib es a melting corner of a s e mi-infinite crystal (Figure 1). The crystal is made of unit cube s , which are initially placed at regula r p ositions and fills the po sitive o ctant x, y , z ≥ 0 of the three dimensional E uclidean space. As the c rystal melts, a finite n umber of cub es a re remov ed from the corner. The pr e sent mode l e xcludes such crystals that hav e “overhangs” viewed from the (1 , 1 , 1) directio n. In other words, the complement o f the crys tal in the p ositive o ctan t is a ssumed to b e a 3 D analogue of Y oung diagrams (Figure2). Since 3D Y oung dia g rams a re represented by “plane partitions” , the melting crystal mo del is a ls o re fer red to as a mo del of “r andom plane partitions” . Though combinatorics of plane partitions has a rather long history [1], Ok- ounko v and Reshetikhin [2] pr op osed an entirely new approach in the course of their study on a kind of sto chastic pr o cess of random par titions (the Sch ur pr o- cess). Their appro ach was based o n “dia g onal slices” o f 3D Y o ung diagr ams and “transfer matr ic e s” b etw een those s lices. As a byproduct, they could re-derive a cla ssical result of MacMa hon [1 ] o n the gener ating function o f the num b ers of plane partitions . Actually , this g enerating function is nothing but the par- tition function o f the a forementioned melting crystal mo del. The metho d of Okounk ov and Reshetikhin was so on g eneralized [3] to dea l with the topolo gical vertex [4 , 5] of A - mo del top ologica l strings on toric Calabi-Y au threefolds. The melting crystal mo del is also closely r elated to supe r symmetric ga uge theories. Namely , with slightest mo dification, the partition function can b e int er preted as the instanton sum o f 5 D N = 1 super symmetric (SUSY) U (1) Y ang- Mills theory on pa rtially compac tified s pace-time R 4 × S 1 [6]. This in- 2 Figure 2 : 3D Y oung diagr am as co mplement of crysta l corner stanton sum is a 5D a nalogue of Nekrasov’s instanton sum for 4D N = 2 SUSY gauge theories [7 , 8]. The 4 D instanton sum is a statistical sum ov er ordinar y partitions (o r “co lored” par titions in the case o f S U ( N ) theory), hence a mo del of rando m partitions. Nekr asov and Okounk ov [9] used such models of r andom partitions to r e -derive the Seib er g-Witten so lutions [10] of 4D N = 2 SUSY gauge theo r ies. Actually , by the afor ementioned metho d of transfer matrices , the sta tistical sum over plane partitions can be reor g anized to a s um ov er par - titions. This is a k ind of q -deforma tions of 4D instanton sums. A 5 D ana logue of the Seibe r g-Witten solution can be derived from this q -defor med instanton sum [9, 11]. In this pa pe r, we review o ur recent results [1 2] on an int eg rable structur e of the melting crystal mo del (and the 5D U (1) insta nt o n sum) with external po tent ia ls. The par tition function Z p ( t ) of this mo del is a function of the cou- pling constants t = ( t 1 , t 2 , . . . ) of the external p otentials. A main conclusio n of these res ults is that Z p ( t ) is, up to a simple fa c to r, a tau function of the 1D T o da hiera r ch y , in other words, a tau function τ p ( t, ¯ t ) of the 2D T o da hierarch y [13] that dep ends only on the difference t − ¯ t of the tw o s ets t, ¯ t of time v a ri- ables. T o der ive this co nclusion, we first rewrite Z p ( t ) in terms o f a complex fermion system. In the c a se of 4D insta n to n sum, such a fermionic repr esenta- tion was pr op osed by Nekrasov et a l. [14, 9]. In the present ca s e, we can use the afore men tio ned transfer matrices [2 ] to construct a fermio nic repr esentation. This fermionic repr e sentation, howev er, do es not take the for m of a sta ndard fermionic repr esentation of the (1D or 2D) T o da hier arch y [15, 1 6]. T o resolve this pro blem, we der ive a set of algebraic relations (r eferred to as “shift sym- metry”) satisfied b y the transfer matrices and a set of fer mion bilinear forms. 3 (Actually , these fermion bilinea r forms turn out to g ive a realiza tion o f “ quan- tum torus L ie algebra”.) T hes e algebraic rela tio ns enable us to rewrite the fermionic r e pr esentation o f Z p ( t ) to the s tandard for m of T o da tau functions. In the 4 D case, a similar pa rtition function with external p otentials has bee n studied by Ma rshakov and Nekrasov [17, 18]. Acco rding to their results, the 1D T o da hierar ch y is also a r elev ant integrable structure therein. Unfortunately , our metho d develop ed for the 5D case relies heavily o n the structure of quantum torus Lie algebr a, which cea s es to exist in the 4D s etup. W e sha ll return to this issue, alo ng with some other iss ue, in the end o f this pap er. This pap er is org anized as follows. Section 2 is a br ief review of the melt- ing crystal mo del and its mathematical background. Section 3 pres ent s the fermionic formula of the partition function. The metho d of transfer matrice s is reviewed in detail. Section 4 deals with the q uantum torus Lie algebra and its shift symmetries. In Section 5, we use this sy mmetr y to rewrite the fermionic representation of the partition function to the sta ndard form a s a T o da tau function. Section 6 is devoted to concluding remark s . 2 Melting crystal mo del 2.1 Y oung diagrams and partitions Let us recall [19] that an ordinary 2 D Y oung dia gram is represented by a n int eg er partition, namely , a sequence λ = ( λ 1 , λ 2 , . . . ) , λ 1 ≥ λ 2 ≥ · · · , of no nincreasing in teger s λ i ∈ Z ≥ 0 with only a finite num b er of λ i ’s b eing nonzero. λ i is the length of i -th r ow of the Y oung dia gram viewed as a collection of unit sq uares. W e shall alwa ys identify suc h a partition λ with a Y oung diagram. The total area o f the diagr a m is given by the degr ee | λ | = X i λ i of the partition. It was shown by E uler that the gener a ting function of the nu mber p ( N ) of partitions λ of degree N has an infinite pro duct formula: ∞ X N =0 p ( N ) q N = n Y n =1 (1 − q n ) − 1 , (2.1) where q is assumed to b e in the range 0 < q < 1. One can interpret this generating function a s the pa rtition function of a mo del o f statistical mechanics, Z 2D = ∞ X N =0 p ( N ) q N = X λ q | λ | , in which ea ch partition λ is assigned a n energy pr o p ortional to | λ | , and q is related to the temper ature T as q = e − const ./T 4 2.2 3D Y oung diagrams and plane partitions A 3D Y oung diagra m ca n b e represented by a “plane partition”, namely , a 2D array π = ( π ij ) ∞ i,j =1 =    π 11 π 12 · · · π 21 π 22 · · · . . . . . . . . .    of nonnega tive in teger s π ij ∈ Z ≥ 0 such that π ij ≥ π i,j +1 , π ij ≥ π i +1 ,j . π ij is the height of the stack of cub e s placed at the ( i, j )-th p ositio n of the plane. W e sha ll identify such a plane partition with the co rresp onding 3D Y oung diagra m. The tota l volume of the 3D Y oung dia gram is given b y | π | = ∞ X i,j =1 π ij . As an ana logue of p ( N ) , one can consider the num b er pp( N ) of pla ne pa r- titions π with | π | = N . The genera ting function of thes e num b ers was studied by Ma cMahon [1] and shown to b e given, again, by an infinite pro duct: ∞ X N =0 pp( N ) q N = ∞ Y n =1 (1 − q n ) − n . (2.2) The right hand side is now called the MacMahon function. In s tatistical me- chanics, this genera ting function b ecomes the par tition function Z 3D = ∞ X N =0 pp( N ) q N = X π q | π | of a canonical ensemble of plane partitions, in which each plane par tition π has an energ y prop ortio nal to the volume | π | . W e shall deform this simplest mo del by external p otentials. T o this end, we hav e to introduce the notio n of “diago nal slices” o f a plane par tition. 2.3 Diagonal slices of 3D Y oung diagrams Given a plane partitio n π = ( π ij ) ∞ i,j =1 , the partition π ( m ) =  ( π i,i + m ) ∞ i =1 if m ≥ 0 ( π j − m,j ) ∞ j =1 if m < 0 is called the m -th diago nal slice of π . These pa rtitions { π ( m ) } ∞ m = −∞ represent a sequence of 2D Y oung diagr a ms that ar e literally o btained by slicing the 3D Y oung diagra ms (Figure3). 5 Figure 3 : Diagona l s lices (b) of plane partition (a ) The dia gonal slices are not ar bitrary but satisfy the co ndition [2, 3] · · · ≺ π ( − 2) ≺ π ( − 1 ) ≺ π (0) ≻ π (1) ≻ π (2) ≻ · · · , (2.3) where “ ≻ ” denotes interlacing r elation , namely , λ = ( λ 1 , λ 2 , . . . ) ≻ µ = ( µ 1 , µ 2 , . . . ) def ⇐ ⇒ λ 1 ≥ µ 1 ≥ λ 2 ≥ µ 2 ≥ · · · . Because of these in ter lacing relatio ns, a pa ir ( T , T ′ ) of s emi-standard tableaux is obtained o n the main diagonal slice λ = π (0) by putting “ m + 1” in b oxes o f the skew diag r am π ( ± m ) /π ( ± ( m + 1)). By this mapping π 7→ ( T , T ′ ), the par titio n function Z 3D of the plane par- titions can be conv erted to a triple sum ov er the tableau T , T ′ and their shap e λ : Z 3D = X λ X T ,T ′ :shap e λ q T q T ′ , (2.4) where q T = ∞ Y m =0 q ( m +1 / 2) | π ( − m ) /π ( − m − 1) | , q T ′ = ∞ Y m =0 q ( m +1 / 2) | π ( m ) /π ( m +1) | . By the well known co m bina torial de finitio n of the Sc hur functions [19], the partial sum ov er the semi-sta ndard tableaux turn o ut to be a sp ecial v alue of 6 the Sch ur functions: X T :shap e λ q T = X T ′ :shap e λ q T ′ = s λ ( q ρ ) , (2.5) where q ρ = ( q 1 / 2 , q 3 / 2 , . . . , q n +1 / 2 , · · · ) . Thu s the partition function can b e even tually r ewritten as Z 3D = X λ s λ ( q ρ ) 2 . (2.6) Let us no te that the specia l v alue of the Sch ur functions has the so called Ho o k formula [1 9] s λ ( q ρ ) = q n ( λ )+ | λ | / 2 Y ( i,j ) ∈ λ (1 − q h ( i,j ) ) − 1 , (2.7) where ( i, j ) stands for the ( i , j )-th b ox in the Y oung diagra m, a nd n ( λ ) is given by n ( λ ) = ∞ X i =1 ( i − 1) λ i . 2.4 Melting crystal model with external p oten tials W e no w defor m the foregoing melting cr ystal mo del b y in tr o ducing the externa l po tent ia ls Φ k ( λ, p ) = ∞ X i =1 q k ( p + λ i − i +1) − ∞ X i =1 q k ( − i +1) with co upling constants t k , k = 1 , 2 , 3 , . . . , on the main diagonal slice λ = π (0 ). The right hand side of the definition of Φ k ( λ, p ) is understo o d to b e a finite sum (hence a rational function of q ) by c a ncellation of terms b etw een the tw o sums: Φ k ( λ, p ) = ∞ X i =1 ( q k ( p + λ i − i +1) − q k ( p − i +1) ) + q k 1 − q pk 1 − q k . The partition function of the deformed mo del reads Z p ( t ) = X π q | π | e Φ( t,π (0) ,p ) , (2.8) where Φ( t, λ, p ) = ∞ X k =1 t k Φ k ( λ, p ) . 7 W e ca n r ep eat the pr evious calculatio ns in this setting to r ewrite the new par- tition function Z p ( t ) as Z p ( t ) = X λ s λ ( q ρ ) 2 q Φ( t,λ,p ) . (2.9) Mo difying this partition function slightly , we obtain the instanton sum of 5D N = 1 SUSY U (1) Y ang-Mills theory [6 ]: Z p ( t ) = X π q | π | Q π (0) e Φ( t,π (0) ,p ) = X λ s λ ( q ρ ) 2 Q | λ | e Φ( t,λ,p ) . (2.10) q and Q are related to physical parameter s R, Λ , ~ of 5D Y ang- Mills theory as q = e − R ~ , Q = ( R Λ) 2 . The external po tent ia ls repr esent the contribution o f Wilson lo ops alo ng the fifth dimension [20]. In this sense, Z p ( t ) / Z p (0) is a gener ating function of corr elation functions o f those Wilson lo op op er ators. Our goa l is to show that the partition function Z p ( t ) is, up to a simple factor, the tau function of the (1 D) T o da hier a rch y . T o this end, we now conside r a fermionic r e pr esentation o f this partition function. 3 F ermionic form u la of p artition function 3.1 Complex fermion system Let ψ ( z ) and ψ ∗ ( z ) denote complex 2 D fermio n fields ψ ( z ) = ∞ X m = −∞ ψ m z − m − 1 , ψ ∗ ( z ) = ∞ X m = −∞ ψ ∗ m z − m . The F ourie r mo des ψ m and ψ ∗ m of ψ ( z ) and ψ ∗ ( z ) sa tisfy the anti-comm utation relations { ψ m , ψ ∗ n } = δ m + n, 0 , { ψ m , ψ n } = { ψ ∗ m , ψ ∗ n } = 0 . The F o ck space F splits into charge p subspa ces F p : F = ∞ M p = −∞ F p . The c har ge p subspace F p has a unique normalized ground state (c har g e p v ac uum) | p i a nd a n orthono rmal basis | λ ; p i labe led by partitions λ . | p i is characterized b y the v a cuum condition ψ m | p i = 0 for m ≥ − p, ψ ∗ m | p i = 0 for m ≥ p + 1 . 8 If the partition is of the form λ = ( λ 1 , . . . , λ n , 0 , 0 , . . . ), the ass o ciated element | λ ; p i o f the basis is o btained from | p i by the action o f fer mio n op era tors as | λ ; p i = ψ − ( p + λ 1 − 1) − 1 · · · ψ − ( p + λ n − n ) − 1 ψ ∗ ( p − n )+1 · · · ψ ∗ ( p − 1)+1 | p i . They a re orthonor ma l in the sense tha t their inner pro ducts hav e the no r malized v alue s h λ ; p | µ ; q i = δ pq δ λµ . 3.2 U (1) curr en t and fermionic represen tation of tau func- tion The U (1) curr e n t J ( z ) o f the complex fermion s y stem is defined as J ( z ) = : ψ ( z ) ψ ∗ ( z ): = ∞ X k = −∞ J m z − m − 1 , where : : denotes the nor mal ordering with resp ect to the v ac uum 0 i : : ψ m ψ ∗ n : = ψ m ψ ∗ n − h 0 | ψ m ψ ∗ n | 0 i . The F ourier mo des J m = ∞ X n = −∞ : ψ m − n ψ ∗ n : of J ( z ) sa tis fy the commutation relations [ J m , J n ] = mδ m + n, 0 (3.1) of the A ∞ Heisenberg algebra, a nd play the role o f “ Hamiltonians” in the usual fermionic formula of the K P and 2 D T o da hie r archies [1 5, 16]. F or the c a se of tau functions τ ( t, ¯ t ), t = ( t 1 , t 2 , . . . ), ¯ t = ( ¯ t 1 , ¯ t 2 , . . . ), of the 2 D T o da hierar ch y , the fermio nic formula reads τ p ( t, ¯ t ) = h p | exp( ∞ X m =1 t m J m ) g exp( − ∞ X m =1 ¯ t m J − m ) | p i (3.2) where g is an element of the infinite dimensio nal Clifford g roup GL ( ∞ ). 3.3 F ermionic represen tation of Z p ( t ) The partition function Z p ( t ) of the deformed melting crystal has a fer mionic representation of the form Z p ( t ) = h p | G + e H ( t ) G − | p i . (3.3) 9 Let us explain the constituents o f this formula a lo ng with an outline of the deriv ation of this fo r mula. H ( t ) is the linea r co mbination H ( t ) = ∞ X k =1 t k H k of the “Hamiltonians” H k = ∞ X n = −∞ q kn : ψ − n ψ ∗ n : . The a forementioned basis element s | λ ; p i of the F ermion F o ck space tur n out to be eig env ectors of these Hamiltonians. The eige n v alues are nothing but the the po tent ia l functions Φ k ( λ, p ): H k | λ ; p i = Φ k ( λ, p ) | λ ; p i . (3.4) G ± are GL ( ∞ ) elements of the s pec ia l form G ± = exp  ∞ X k =1 q k/ 2 k (1 − q k ) J ± k  . Since the numerical factors q k/ 2 / (1 − q k ) in this definition can b e expanded as q k/ 2 1 − q k = − 1 X m = −∞ q − k ( m +1 / 2) = ∞ X m =0 q k ( m +1 / 2) , one can factorize these op er ators as G + = − 1 Y m = −∞ Γ + ( m ) , G − = ∞ Y m =0 Γ − ( m ) , (3.5) where Γ ± ( m ) = e x p  ∞ X k =1 1 k q ∓ k ( m +1 / 2) J ± k  . These Γ ± ( m )’s a re a sp ecializatio n of the so called vertex o p e rators V ± ( z ) = ex p  ∞ X k =1 z k k J ± k  for b oso nization of the complex fermio ns. F ollowing the idea of Okounko v and Reshetikhin [2], we now cons ide r m as a fictitious “time” v aria ble. A plane par titio n then may b e thought of as the 10 “path” (or “ world volume”) of discrete time evolutions o f a par tition λ that starts from the e mpt y partition ∅ = (0 , 0 , . . . ) at infinite past a nd ends aga in in ∅ at infinite future (see Figure3). The vertex o per ators Γ ± ( m ) play the ro le o f transfer matrice s b etw een neighboring dia gonal slices . The vertex o p erators Γ ± ( m ) act o n the afor ement io ned o rthonormal bases | λ ; p i a nd h λ ; p | as h λ ; p | Γ + ( m ) = X µ ≻ λ h µ ; p | q − ( m +1 / 2)( | µ |−| λ | ) (3.6) for m = − 1 , − 2 , . . . a nd Γ − ( m ) | λ ; p i = X µ ≻ λ q ( m +1 / 2)( | µ |−| λ | ) | µ ; p i (3.7) for m = 0 , 1 , . . . [2, 3]. The rig ht ha nd side of these for mu la s give a linear combination of all p o ssible time e volutions of the m -th slice λ = π ( m ) at the next time. The weigh t q ∓ ( m +1 / 2)( | µ |−| λ | ) of each state o n the rig ht hand side are e x actly the factor s ass igned to the boxes o f π ( m ) /π ( m ∓ 1) in the definition of the weigh ts q T , q T ′ that a ppea r in the co mbinatorial formula (2.4). Since G ± are pro ducts of these slice-to -slice “tra nsfer matrices”, h p | G + and G − | p i b ecome linear combinations of the states h λ ; p | and | λ ; p i that evolv e from the ground states h p | and | p i a t m = ∓∞ . By what we hav e s een a b ove, the weigh ts of h λ ; p | and | λ ; p i in these linear combinations ar e given b y the partia l sums of q T and q T ′ ov er a ll semi-standar d tablea ux T and T ′ of shap e λ , namely , the sp ecial v alue s λ ( q ρ ) of the Sch ur function. Th us h p | G + and G − | p i can be expressed as h p | G + = X λ X T :shap e λ q T h λ ; p | = X λ s λ ( q ρ ) h λ ; p | , (3.8) G − | p i = X λ X T ′ :shap e λ q T ′ | λ ; p i = X λ s λ ( q ρ ) | λ ; p i . (3.9) The exp ectation v a lue of e H ( t ) with resp ect to these states yields the fermionic representation (3.3) of the pa r tition function Z p ( t ). The fermionic re pr esentation (3.3) is apparently different from the fermionic formula (3.2) of ta u functions of the 2D T o da hier arch y . T o show that Z p ( t ) is indeed a tau function, we hav e to rewrite (3.3) to the form o f (3.2). This is the place wher e the quantum torus Lie algebr a joins the game. 11 4 Quan tum torus Lie algebra 4.1 F ermionic realization of quantum tor us Lie algebra Let V ( k ) m ( k = 0 , 1 , . . . , m ∈ Z ) denote the following fermion bilinear forms: V ( k ) m = q − km/ 2 ∞ X n = −∞ q kn : ψ m − n ψ ∗ n : = q k/ 2 I dz 2 π i z m : ψ ( q k/ 2 z ) ψ ∗ ( q − k/ 2 z ): Note that J m = V (0) m , H k = V ( k ) 0 . Actually , V ( k ) m coincides with O kounk ov and Pandharipande’s op erator E m ( z ) [21, 22] sp ecialized to z = q k . As they found fo r E m ( z ), our V ( k ) m ’s satisfy the commutation relations [ V ( k ) m , V ( l ) n ] = ( q ( lm − kn ) / 2 − q ( kn − lm ) / 2 )( V ( k + l ) m + n − δ m + n, 0 q k + l 1 − q k + l ) . (4.1) This is a (central extension of ) q -defor mation of the Poisson algebr a of functions on a 2 -torus. W e refer to this Lie algebra as “ quantum tor us Lie alg ebra”. Mo re precisely , a full quantum torus Lie algebra should c ontain elements for k < 0 as well; fo r several reas ons, we shall not include those ele men ts. 4.2 Shift symmetry among basis of quan tum torus Lie al- gebras The following rela tions, which we call “shift s ymmetry”, play a central role in ident ifying Z p ( t ) as a ta u function: G − G +  V ( k ) m − δ m, 0 q k 1 − q k  ( G − G + ) − 1 = ( − 1 ) k  V ( k ) m + k − δ m + k, 0 q k 1 − q k  (4.2) These rela tions are derived as follows. Let us recall that the fermion fields ψ ( z ) , ψ ∗ ( z ) transform under adjoint action by J ± k ’s as exp  ∞ X k =1 c k J ± k  ψ ( z ) exp  − ∞ X k =1 c k J ± k  = exp  ∞ X k =1 c k z ± k  ψ ( z ) , ( 4 .3) exp  ∞ X k =1 c k J ± k  ψ ∗ ( z ) exp  − ∞ X k =1 c k J ± k  = exp  − ∞ X k =1 c k z ± k  ψ ∗ ( z ) . (4.4) 12 By letting c k = q k/ 2 / (1 − q k ), the exp onential op e r ators in these formulas turn int o G ± , s o that we hav e the op era to r iden tities G + ψ ( z ) G + − 1 = ( q 1 / 2 z ; q ) − 1 ∞ ψ ( z ) , (4.5) G + ψ ∗ ( z ) G + − 1 = ( q 1 / 2 z ; q ) ∞ ψ ∗ ( z ) , (4.6) G − ψ ( z ) G − − 1 = ( q 1 / 2 z − 1 ; q ) − 1 ∞ ψ ( z ) , (4.7) G − ψ ∗ ( z ) G − − 1 = ( q 1 / 2 z − 1 ; q ) ∞ ψ ∗ ( z ) , (4.8) where ( z ; q ) ∞ denotes the standar d q -factor ia l symbol ( z ; q ) ∞ = ∞ Y n =0 (1 − z q n ) . W e use these op erator iden tities to der ive transformation of the fermio n bilinear for ms : ψ ( q k/ 2 z ) ψ ∗ ( q − k/ 2 z ): under conjuga tion by G ± . Since : ψ ( q k/ 2 z ) ψ ∗ ( q − k/ 2 z ): = − ψ ∗ ( q − k/ 2 z ) ψ ( q k/ 2 z ) + q k/ 2 (1 − q k ) z , let us first consider ψ ∗ ( q − k/ 2 z ) ψ ( q k/ 2 z ). Under conjugation by G + , it trans- forms a s G + ψ ∗ ( q − k/ 2 z ) ψ ( q k/ 2 z ) G + − 1 = ( q 1 / 2 · q − k/ 2 z ; q ) ∞ ( q 1 / 2 · q k/ 2 z ; q ) ∞ ψ ∗ ( q − k/ 2 z ) ψ ( q k/ 2 z ) = k Y m =1 (1 − z q ( k +1) / 2 − m ) ψ ∗ ( q − k/ 2 z ) ψ ( q k/ 2 z ) . This implies that G +  : ψ ( q k/ 2 z ) ψ ∗ ( q − k/ 2 z ): − q k/ 2 (1 − q k ) z  G + − 1 = k Y m =1 (1 − q ( k +1) / 2 − m z )  : ψ ( q k/ 2 z ) ψ ∗ ( q − k/ 2 z ): − q k/ 2 (1 − q k ) z  . (4.9 ) In m uch the sa me wa y , we c an der ive a similar transformatio n under conjuga tion by G − . In this case, it is more conv enient to rewr ite the result a s follows: G − − 1  : ψ ( q k/ 2 z ) ψ ∗ ( q − k/ 2 z ): − q k/ 2 (1 − q k ) z  G − = k Y m =1 (1 − q − ( k +1) / 2+ m z − 1 )  : ψ ( q k/ 2 z ) ψ ∗ ( q − k/ 2 z ): − q k/ 2 (1 − q k ) z  . (4.10) 13 W e note here that the pr efactors on the right hand side of the last t wo equations ar e related as k Y m =1 (1 − q ( k +1) / 2 − m z ) = ( − z ) k k Y m =1 (1 − q − ( k +1) / 2+ m z − 1 ) . Accounting for this simple, but sig nificant relation, we can der ive the identit y G − G +  : ψ ( q k/ 2 z ) ψ ∗ ( q − k/ 2 z ): − q k/ 2 (1 − q k ) z  ( G + G − ) − 1 = ( − z ) k  : ψ ( q k/ 2 z ) ψ ∗ ( q − k/ 2 z ): − q k/ 2 (1 − q k ) z  . (4.11) The shift symmetry (4.2) fo llows immediately from this identit y . When m = 0 and m = − k , (4.2 ) ta kes the particula r form G − G +  V ( k ) 0 − q k 1 − q k  ( G − G + ) − 1 = ( − 1) k V ( k ) k , (4.12) ( G − G + ) − 1  V ( k ) 0 − q k 1 − q k  G − G + = ( − 1) k V ( k ) − k . (4.13) It is these identities that we shall use to con vert the fermionic representation of Z p ( t ) to the s tandard fer mionic formula of tau functions. 5 In tegrable structure of melting crystal mo del 5.1 P artit ion function as tau function of 2D T o da hierar- c hy Let us split the o pe rator G + e H ( t ) G − in (3.3) into three pieces as G + e H ( t ) G − = G + e H ( t ) / 2 e H ( t ) / 2 G − = G + e H ( t ) / 2 G + − 1 · G + G − · G − − 1 e H ( t ) / 2 G − and use the sp ecial cas es (4.12) and (4.13) of the shift symmetry to r ewrite those pieces. T o this end, it is conv e nie nt to r ewrite (4.1 2) and (4.13) a s G +  H k − q k 1 − q k  G + − 1 = ( − 1) k G − − 1 V ( k ) k G − , G − − 1  H k − q k 1 − q k  G − = ( − 1) k G + V ( k ) − k G + − 1 . Though the op era to rs V ( k ) ± k on the right hand side a r e unfamiliar in the theor y of integrable hierarchies, we can conv ert them to the familia r “ Ha miltonians” J ± k = V (0) ± k of the T o da hierar chies as q W/ 2 V ( k ) k q − W/ 2 = V (0) k = J k , q − W/ 2 V ( k ) − k q W/ 2 = V (0) − k = J − k , (5.1) 14 where W is a sp ecial element of W ∞ algebra: W = W (3) 0 = ∞ X n = −∞ n 2 : ψ − n ψ ∗ n : W e thus even tually obta in the relations G +  H k − q k 1 − q k  G + − 1 = ( − 1) k G − − 1 q − W/ 2 J k q W/ 2 G − , (5.2) G − − 1  H k − q k 1 − q k  G − = ( − 1) k G + q W/ 2 J − k q − W/ 2 G + − 1 (5.3) betw e e n H k ’s and J ± k ’s. By these rela tions, G + e H ( t ) / 2 G + − 1 can b e calculated as G + e H ( t ) / 2 G + − 1 = exp  ∞ X k =1 t k q k 2(1 − q k )  G − − 1 q − W/ 2 exp  ∞ X k =1 ( − 1) k t k 2 J k  q W/ 2 G − . A similar expressio n ca n b e derived for G − − 1 e H ( t ) G − as well. W e can thus rewrite G + e H ( t ) G − as G + e H ( t ) G − = exp  ∞ X k =1 t k q k 1 − q k  G − − 1 q − W/ 2 exp  ∞ X k =1 ( − 1) k t k 2 J k  × × g exp  ∞ X k =1 ( − 1) k t k 2 J − k  q − W/ 2 G + − 1 where g = q W/ 2 ( G − G + ) 2 q W/ 2 . (5.4) The partition function Z p ( t ) is g iven by the expec tation v alue of this op erator with resp ect to h p | and | p i . Since the a c tion by the leftmost and r ightmost pieces of g yie lds only a sca lar multiplier to h p | , | p i as h p | G − − 1 q − W/ 2 = q − p ( p +1)(2 p +1) / 12 h p | , (5.5) q − W/ 2 G + − 1 | p i = q − p ( p +1)(2 p +1) / 12 | p i , (5.6) Z p ( t ) can b e expressed as Z p ( t ) = exp  ∞ X k =1 t k q k 1 − q k  q − p ( p +1)(2 p +1) / 6 × × h p | exp  ∞ X k =1 ( − 1) k t k 2 J k  g exp  ∞ X k =1 ( − 1) k t k 2 J − k  | p i . (5.7 ) The exp ectation v alue h p | · · · | p i takes exactly the for m o f (3.2). Thus, up to the simple prefa ctor, Z p ( t ) is essentially a tau function of the 2 D T o da hierarch y . Thu s we find that an integrable structure be hind the melting crystal mo del is the 2 D T o da hiera r ch y . This is , howev er, not the end of the stor y . 15 5.2 1D T o da hierarch y as true integrable str ucture The foreg oing calculatio n is ba sed on the splitting G + e H ( t ) G − = G + e H ( t ) / 2 G + − 1 · G + G − · G − − 1 e H ( t ) / 2 G − . Actually , we could have started from a different splitting of G + e H ( t ) G − , e.g., G + e H ( t ) G − = G + e H ( t ) G + − 1 · G + G − = G + G − · G − − 1 e H ( t ) G − This leads to another set of expres sions of Z p ( t ) in which only the h p | · · · | p i part is differ ent . W e thus hav e the following thre e differ e nt expr essions for this part: h p | exp  ∞ X k =1 ( − 1) k t k 2 J k  g exp  ∞ X k =1 ( − 1) k t k 2 J − k  | p i = h p | exp  ∞ X k =1 ( − 1) k t k J k  g | p i = h p | g exp  ∞ X k =1 ( − 1) k t k J − k  | p i . (5.8) These identities of the exp ectation v alues can b e dir ectly der ived from the op erator identities J k g = g J − k , k = 1 , 2 , 3 , . . . (5.9) satisfied by g . (These o p er ator identities themselves are a consequences o f the shift symmetry of V ( k ) m ’s.) Gener a lly s pe aking, this kind of op er ator identities imply symmetry constr a ins o n the tau functions [23, 24]; in the present case, the co nstraints r ead ∂ ∂ t k τ p ( t, ¯ t ) + ∂ ∂ ¯ t k τ p ( t, ¯ t ) = 0 , k = 1 , 2 , 3 , . . . . (5.10) In o ther words, the ta u function is a function of t − ¯ t , τ p ( t, ¯ t ) = τ p ( t − ¯ t, 0) = τ p (0 , ¯ t − t ) , and reduces to a tau function τ p ( t ) of the 1D T o da hierarchy tha t has a single series of time v ariables t = ( t 1 , t 2 , . . . ) r ather than the tw o ser ies of the 2D T o da hierarch y . Thus the 1D T o da hiera rch y even tually turns out to b e an underlying int eg rable structure of the deformed melting crys tal mo del. The same conclusion ca n b e der ived for the instanton sum (2.1 0) of 5D SUSY U (1) Y ang-Mills theory . It has a fermionic r epresentation of the fo r m Z p ( t ) = h p | G + Q L 0 e H ( t ) G − | p i (5.11) where L 0 is a sp ecial element of the Viraso ro algebr a: L 0 = ∞ X n = −∞ n : ψ − n ψ ∗ n : . 16 One can r epe at almost the s ame calculations as the previous case to conv er t Z p ( t ) to the fo r m o f (5.7). The c o unterpart of g is g iven b y g = q W/ 2 G − G + Q L 0 G − G + q W/ 2 , (5.12) which, to o, sa tisfy the reduction conditions (5.9) to the 1D T o da hier arch y . Thu s a relev ant integrable structur e is aga in the 1D T o da hier arch y . 6 Concluding remarks 6.1 Problems on 4D instan ton sum In der iving the instan ton sum (2.10), 5D spa c e-time is par tia lly compactified in the fifth dimension as R 4 × S 1 . The parameter R is the r adius of S 1 . The r efore, letting R → 0 amo unt s to 4 D limit. Unfortunately , it is not straig ht for ward to achieve such a 4D limit in the present setup. Firstly , the 5D instanton sum with exter nal p o tentials do es no t hav e a r easonable limit as R → 0 . Any naive prescr iption letting R → 0 yields a res ult in which t dep endence disapp ear s or be c o mes trivia l [12]. Seco ndly , the shift symmetr y of the quantum torus Lie a lgebra ceases to exist in the limit as q = e − R ~ → 1. Sp eak ing more pr ecisely , the quantum torus Lie a lgebra itself turns int o a W ∞ algebra in this limit, but no analogue of shift symmetry (4.2) is known for the la tter case. F or these r easons, the 4D ca se has to b e studied independently . The 4D insta nt o n sum [7 , 8], to o, is a sum ov e r partitions . Moreov er , this statistical sum has a fermio nic representation [1 4, 9]. Marsha kov a nd Nekrasov [17, 18] further intro duced external p otentials therein. Actually , the 4D insta n- ton sum for U (1) gauge theo r y is almo st ident ica l to the g enerating function o f Gromov-Witten inv a riants of CP 1 [21, 22]. This can b e most cle a rly seen in the fermionic r e pr esentation o f these genera ting functions, which reads Z 4D p ( t ) = h p | e J 1 / ~ exp  ∞ X k =1 t k P k +1 k + 1  e J − 1 / ~ | p i , (6.1) where P k ’s are fermion bilinear forms introduced by Okounko v and Pandhari- pande fo r a fermionic repre s entation o f (absolute) Gro mov-Witten inv ar ia nts of CP 1 [21]. As r egards these Gromov-Witten in v ar iants (in o ther words, co r- relation functions o f the to p o logical σ mo del) , it ha s b een known for years [25, 26, 27, 28, 2 9] that a relev a nt int eg rable structure is the 1 D T o da hierar ch y . Thu s the 1D T o da hierar ch y is ex p ected to b e the int eg rable structur e of the 4D instanton sum as well. This has b een co nfir med b y Ma rshakov and Nekrasov in detail [17, 18]. What is s till missing, how ever, is a formula like (5.7) that dire c tly connects Z 4D p ( t ) with the standar d fermionic formula (3.2) of the tau function. Finding a 4D analog ue of (5.7) is thus an int r iguing op en problem. This iss ue is a lso closely related to the fate of shift s ymmetry (4.2) in the q → 1 limit. 17 6.2 Relation to t op ological strings Our r esults are directly or indir ectly connected with some a s pe cts of topolo gical strings as well. 1. According to the theo ry of top ologic al vertex [5], the partition function Z p ( t ) of the defor med melting crystal mo de l has another interpretation as the A -mo del top olog ical str ing amplitude for the toric Calabi-Y au threefold O ⊕ O ( − 2) → CP 1 . In this interpretation, q and Q are para metr ized by the str ing coupling co nstant g st and the K¨ ahler volume a of CP 1 as q = e − g st , Q = e − a . Spec ia lizing the v alue of t leads to several interesting obser v a tions [12]. 2. A g e ne r ating function of the t wo-legged topo logical v er tex W λµ ∼ c λµ • is known to give a tau function of the 2D T o da hierar ch y [30]. In the fer mio nic representation (3.2), this amounts to the case where g = q W/ 2 G + G − q W/ 2 . (6.2) (Actually , for complete agree ment with the usual conv ention, we hav e to repla c e W with K = ∞ X n = −∞  n − 1 2  2 : ψ − n ψ ∗ n : , but this is not a s e rious pr oblem. The difference can b e absorb ed by resca ling t k ’s.) Let us s tress that this GL ( ∞ ) element do es not satisfy the reductio n condition (5.9) to the 1D T o da hierarchy . 3. A g e nerating function o f double Hurwitz num b er s for cov er ing s of CP 1 gives yet a nother type of tau function of the 2D T o da hiera rch y [31]. Actually , the GL ( ∞ ) element for the fer mio nic repr esentation is given by g = q W/ 2 . (6.3) (More precis ely , a s in the previous case, W ha s to b e replaced b y K , but the difference is again irrelev ant.) In this case, the reduction co ndition (5.9) to the 1D T o da hierarchy is no t sa tisfied, but the op er ator ident ities (5.1) imply that another set of reduction conditions a re hidden b ehind (see be low). 6.3 Constrain ts and quan t um torus Lie algebra As a conse q uence of the shift symmetry of V ( k ) m ’s, the GL ( ∞ ) elemen ts g of the aforementioned models o f top ologic al strings turn out to satisfy some algebraic relations other than (5.9). According to general results o n co nstraints of the 2D T o da hiera r ch y [23, 24], such relations imply the existence of constra int s on the tau functions a nd the Lax and Orlov-Sc hulman op er ators. Those constr aints inherit the structure of the quantum torus Lie algebra. Let us illustr ate this observ ation for the case o f double Hurwitz num ber s over CP 1 . 18 The GL ( ∞ ) element g = q W/ 2 for this case satisfies the op erator identities J k g = g V ( k ) k , g J − k = V ( k ) − k g (6.4) as a consequence o f (5.1). These ident ities ca n b e conv erted to the constra int s L = q 1 / 2 q ¯ M ¯ L, ¯ L − 1 = q − 1 / 2 q M L − 1 (6.5) on the Lax and Orlov-Sch ulman o pe r ators L, M , ¯ L, ¯ M of the 2D T o da hierarchy . Emergence of the exp onential o p erators q M and q ¯ M is a manifestation of the quantum tor us Lie alg ebra. T o see this, let us r ecall that the Lax and O rlov- Sch ulman op era tors satisfy the (t wisted) cano nic a l commutation relations [ L, M ] = L, [ ¯ L, ¯ M ] = ¯ L. (6.6 ) This implies that the monomials q − km/ 2 L m q kM and q − km/ 2 ¯ L m q k ¯ M of L, q M , ¯ L, q ¯ M give tw o copies of realizations o f the quantum torus Lie algebr a. The co nstraints (6.5) are remark ably simila r to the “ string e q uations” L = ¯ M ¯ L, ¯ L − 1 = M L − 1 (6.7) of c = 1 s trings at self-dual radius [32, 3 3, 23, 2 4]. A relev ant algebr aic structure of these string equa tions is the W ∞ algebra. Thus (6.5) may b e thoug ht of as q -deforma tions of these W -a lgebraic co ns traints. Ac knowledge ments W e are grateful to Nikita Nekras ov and Motohico Mulase for v a luable comments and fr uitful discussion. K .T is partly supp or ted by Grant-in-Aid for Scientific Research No. 1834006 1 and No. 19 54017 9 from the Japan So ciety for the Promotio n of Science. References [1] D.M. Br e ssoud, Pro ofs and c o nfirmations: The sto ry of the a lternating sig n matrix co njecture, Cambridge Univ ers ity Pre s s, 1999 . [2] A. Okounk ov and N. Reshetikhin, Cor relation function of Sch ur Pro cess with application to lo cal g eometry o f a random 3-Dimensiona l young dia- gram, J. Amer. Math. So c. 1 6 , (200 3), 581 –603 . [3] A. Okounk ov, N. Reshetikhin and C. V afa, Qua ntu m Calabi-Y au and clas- sical crystals, in: P . Etingof, V. Retakh and I.M. Singer (eds.), The u nity of mathematics , P rogr . Ma th. 24 4 , Birkh¨ auser, 2006, pp. 59 7–618 . [4] A. Iqbal, All genus top olog ical string amplitudes and 5 -brane webs as F eyn- man dia grams, a rXiv:hep-th/020 7114 . 19 [5] M. Aganagic, A. K lemm, M. Mari ˜ no and C. V a fa , The top olo gical vertex, Commun. Math. P hys. 254 (2005 ), 425 –478 . [6] T. Maeda, T. Nak atsu, K. T a k a s aki and T. T a makoshi, Five-dimensional sup e rsymmetric Y a ng-Mills theories a nd r andom plane partitions, JHEP 0503 (2 0 05), 056 . [7] N.A. Nekrasov, Seib er g -Witten Prep otential fro m Instant o n Counting, Adv. Theo r. Math. Phys. 7 (2004), 83 1–864 . [8] H. Nak a jima and K . Y oshiok a, Instanton counting on blowup, I. 4- dimensional pur e g auge theory , Inv ent. Ma th 16 2 (200 5), 313– 355. [9] N. Nekra sov and A. Okounko v, Seib er g-Witten theo ry and ra ndom par- titions, in: P . E tingof, V. Reta kh and I.M. Singer (eds .), The unity of mathematics , Pro gr. Math. 244 , Bir kh¨ auser, 20 06, pp. 525– 296. [10] N. Seib erg a nd E. Witten, E lectric-mag ne tic duality , mono po le condensa - tion, and confinement in N = 2 sup ersymmetric Y ang-Mills theory , Nucl. Phys. B426 (1994 ) 19– 52; Er ratum, ibid. B430 (19 9 4) 48 5; Monop oles, duality and chiral sy mmetry br eaking in N = 2 Sup ersymmetr ic Q CD, ibid. B43 1 (199 4) 49 4–550 . [11] T. Maeda, T. Nak atsu, K. T ak a saki a nd T. T amakoshi, F ree fermion and Seibe r g-Witten different ia l in rando m plane partitions, Nucl. Phys. B7 15 (2005), 275–3 03. [12] T. Na k atsu and K. T ak asaki, Melting cry s tal, quantum to r us and T o da hierarch y , Commun. Math. P hys. 285 (200 9), 445– 468. [13] K. Ueno a nd K. T ak asaki, T o da lattice hierarch y , Adv. Stud. P ure Math. 4 (1 984), 1– 95. [14] A. Losev, A. Ma rshakov and N. Nekras ov, Small instantons, little strings and free fermions, M. Shifman, A. V ains tein and J. Wheater (eds.), Ian Ko - gan memor ial volume, F r om fields to st rings: cir cu m navigating the or etic al physics , pp. 581 – 621, W orld Scientific, 2005. [15] M. Jimbo and T. Miw a, So litons and infinite dimensional Lie algebr as, Publ. RIMS, Kyoto Univ., 19 (19 83), 9 43–10 01. [16] T. T akebe , Representation theoretica l mea nings of the initial v a lue problem for the T o da la ttice hier arch y I, Lett. Ma th. P hys. 21 (19 91); ditto I I, Publ. RIMS, K yoto Univ., 27 (1991 ), 491–50 3. [17] A. Marshakov and N. Nekra sov, Extended Seib er g-Witten theory and in- tegrable hierarch y , JHEP 0701 (2 0 07), 104 . [18] A. Ma r shako v , On microscopic o rigin o f in teg rability in Seiberg -Witten theory , The o r. Math. Phys. 154 (20 08), 36 2–384 ; Seib erg- Witten theo ry and extended T o da hierar chy , J HE P 0803 (2008), 05 5. 20 [19] I.G. Macdonald, Symmetric functions a nd Hall p olyno mials, O xford Uni- versit y Press, 19 95. [20] L. Baulieu, A. Losev and N.Nekra sov, C her n-Simons and twisted sup er - symmetry in v arious dimensions, Nucl. P hys. B522 (199 8), 82–1 04. [21] A. Okounko v and R. Pandharipande, Gr omov-Witten theo ry , Hurwitz the- ory , and completed cycles, Ann. Math. 163 (2006), 51 7–56 0. [22] A. O kounk ov and R. Pandharipande, The equiv ar iant Gr omov-Witten the- ory of P 1 , Ann. Math. 163 (2006), 56 1–60 5 . [23] T. Nak atsu, K. T ak asa k i and S. Tsujimaru, Quant um and classic a l asp ects of defo rmed c = 1 strings, Nucl. P hys. B44 3 (199 5), 15 5–197 . [24] K. T ak asa ki, T o da la ttice hierarch y and genera lized str ing equa tions, Com- m un. Math. Phys. 181 (1996), 13 1–15 6. [25] T. E guchi, K. Hori and S.-K. Y ang, T op olo gical σ mo dels and la rge-N matrix integral, Int. J. Mo d. Phys. A 10 (199 5 ), 4203– 4224 . [26] T. E g uchi, K. Hori and C.-S. Xiong, Quantum co homology and Viraso ro algebra, P hys. Lett. B402 (1997 ), 71–80 . [27] E. Getzler , The T o da conjecture, in: K. F uk aya et al. (eds.), Symple ctic ge ometry and mirr or symmetry (KIAS, Seoul, 2000 ), W orld Scientific, Sin- gap ore, 2001 , pp. 51 –79. [28] R. Pandharipande, The T o da equations a nd Gromov-Witten theory of the Riemann s phere, Lett. Math. Phys. 53 (2 0 00), 59– 74. [29] A. Giv ental, Gromov-Witten inv aria nts and quantization of quadra tic Hamiltonians, Moscow Math. J. 1 (20 01), 5 51–5 6 8. [30] J. Zhou, Ho dge in teg rals and integrable hierarchies, arXiv:math.AG/031040 8. [31] A. Okounk ov, T o da equations for Hurwitz num b ers, Math. Res. Lett. 7 (2000), 447–4 53. [32] T. Eg uchi and H. Kanno , T o da la ttice hierar chy a nd the top ologica l de- scription o f c = 1 string theory , P hys. Lett. B331 (1994), 330– 334. [33] K. T ak asa ki, Disp ersionless T o da hiera rch y a nd tw o-dimensio nal s tring the- ory , Commun. Math. Phys. 170 , 10 1–11 6 . 21