Integrable structure of melting crystal model with external potentials
aa r X i v : . [ m a t h - ph ] F e b Integrable structure of melting crystal modelwith external potentials
Toshio Nakatsu and Kanehisa Takasaki ∗ Faculty of Engineering, Mathematics and Physics,Setsunan UniversityIkedanakamachi, Neyagawa, Osaka 572-8508, Japan Graduate School of Human and Environmental Studies, Kyoto UniversityYoshida, Sakyo, Kyoto 606-8501, Japan
Abstract
This is a review of the authors’ recent results on an integrable structureof the melting crystal model with external potentials. The partition func-tion of this model is a sum over all plane partitions (3D Young diagrams).By the method of transfer matrices, this sum turns into a sum over ordi-nary partitions (Young diagrams), which may be thought of as a modelof q -deformed random partitions. This model can be further translatedto the language of a complex fermion system. A fermionic realization ofthe quantum torus Lie algebra is shown to underlie therein. With theaid of hidden symmetry of this Lie algebra, the partition function of themelting crystal model turns out to coincide, up to a simple factor, with atau function of the 1D Toda hierarchy. Some related issues on 4D and 5Dsupersymmetric Yang-Mills theories, topological strings and the 2D Todahierarchy are briefly discussed. ∗ E-mail: [email protected]
The melting crystal model is a model of statistical mechanics that describes amelting corner of a semi-infinite crystal (Figure1). The crystal is made of unitcubes, which are initially placed at regular positions and fills the positive octant x, y, z ≥ , ,
1) direction. In otherwords, the complement of the crystal in the positive octant is assumed to bea 3D analogue of Young diagrams (Figure2). Since 3D Young diagrams arerepresented by “plane partitions”, the melting crystal model is also referred toas a model of “random plane partitions”.Though combinatorics of plane partitions has a rather long history [1], Ok-ounkov and Reshetikhin [2] proposed an entirely new approach in the course oftheir study on a kind of stochastic process of random partitions (the Schur pro-cess). Their approach was based on “diagonal slices” of 3D Young diagrams and“transfer matrices” between those slices. As a byproduct, they could re-derivea classical result of MacMahon [1] on the generating function of the numbersof plane partitions. Actually, this generating function is nothing but the par-tition function of the aforementioned melting crystal model. The method ofOkounkov and Reshetikhin was soon generalized [3] to deal with the topologicalvertex [4, 5] of A -model topological strings on toric Calabi-Yau threefolds.The melting crystal model is also closely related to supersymmetric gaugetheories. Namely, with slightest modification, the partition function can beinterpreted as the instanton sum of 5D N = 1 supersymmetric (SUSY) U (1)Yang-Mills theory on partially compactified space-time R × S [6]. This in-2igure 2: 3D Young diagram as complement of crystal cornerstanton sum is a 5D analogue of Nekrasov’s instanton sum for 4D N = 2 SUSYgauge theories [7, 8]. The 4D instanton sum is a statistical sum over ordinarypartitions (or “colored” partitions in the case of SU ( N ) theory), hence a modelof random partitions. Nekrasov and Okounkov [9] used such models of randompartitions to re-derive the Seiberg-Witten solutions [10] of 4D N = 2 SUSYgauge theories. Actually, by the aforementioned method of transfer matrices,the statistical sum over plane partitions can be reorganized to a sum over par-titions. This is a kind of q -deformations of 4D instanton sums. A 5D analogueof the Seiberg-Witten solution can be derived from this q -deformed instantonsum [9, 11].In this paper, we review our recent results [12] on an integrable structureof the melting crystal model (and the 5D U (1) instanton sum) with externalpotentials. The partition function Z p ( t ) of this model is a function of the cou-pling constants t = ( t , t , . . . ) of the external potentials. A main conclusion ofthese results is that Z p ( t ) is, up to a simple factor, a tau function of the 1DToda hierarchy, in other words, a tau function τ p ( t, ¯ t ) of the 2D Toda hierarchy[13] that depends only on the difference t − ¯ t of the two sets t, ¯ t of time vari-ables. To derive this conclusion, we first rewrite Z p ( t ) in terms of a complexfermion system. In the case of 4D instanton sum, such a fermionic representa-tion was proposed by Nekrasov et al. [14, 9]. In the present case, we can usethe aforementioned transfer matrices [2] to construct a fermionic representation.This fermionic representation, however, does not take the form of a standardfermionic representation of the (1D or 2D) Toda hierarchy [15, 16]. To resolvethis problem, we derive a set of algebraic relations (referred to as “shift sym-metry”) satisfied by the transfer matrices and a set of fermion bilinear forms.3Actually, these fermion bilinear forms turn out to give a realization of “quan-tum torus Lie algebra”.) These algebraic relations enable us to rewrite thefermionic representation of Z p ( t ) to the standard form of Toda tau functions.In the 4D case, a similar partition function with external potentials has beenstudied by Marshakov and Nekrasov [17, 18]. According to their results, the 1DToda hierarchy is also a relevant integrable structure therein. Unfortunately,our method developed for the 5D case relies heavily on the structure of quantumtorus Lie algebra, which ceases to exist in the 4D setup. We shall return to thisissue, along with some other issue, in the end of this paper.This paper is organized as follows. Section 2 is a brief review of the melt-ing crystal model and its mathematical background. Section 3 presents thefermionic formula of the partition function. The method of transfer matrices isreviewed in detail. Section 4 deals with the quantum torus Lie algebra and itsshift symmetries. In Section 5, we use this symmetry to rewrite the fermionicrepresentation of the partition function to the standard form as a Toda taufunction. Section 6 is devoted to concluding remarks. Let us recall [19] that an ordinary 2D Young diagram is represented by aninteger partition, namely, a sequence λ = ( λ , λ , . . . ) , λ ≥ λ ≥ · · · , of nonincreasing integers λ i ∈ Z ≥ with only a finite number of λ i ’s beingnonzero. λ i is the length of i -th row of the Young diagram viewed as a collectionof unit squares. We shall always identify such a partition λ with a Youngdiagram. The total area of the diagram is given by the degree | λ | = X i λ i of the partition.It was shown by Euler that the generating function of the number p ( N ) ofpartitions λ of degree N has an infinite product formula: ∞ X N =0 p ( N ) q N = n Y n =1 (1 − q n ) − , (2.1)where q is assumed to be in the range 0 < q <
1. One can interpret thisgenerating function as the partition function of a model of statistical mechanics, Z = ∞ X N =0 p ( N ) q N = X λ q | λ | , in which each partition λ is assigned an energy proportional to | λ | , and q isrelated to the temperature T as q = e − const ./T .2 3D Young diagrams and plane partitions A 3D Young diagram can be represented by a “plane partition”, namely, a 2Darray π = ( π ij ) ∞ i,j =1 = π π · · · π π · · · ... ... . . . of nonnegative integers π ij ∈ Z ≥ such that π ij ≥ π i,j +1 , π ij ≥ π i +1 ,j .π ij is the height of the stack of cubes placed at the ( i, j )-th position of theplane. We shall identify such a plane partition with the corresponding 3DYoung diagram. The total volume of the 3D Young diagram is given by | π | = ∞ X i,j =1 π ij . As an analogue of p ( N ), one can consider the number pp( N ) of plane par-titions π with | π | = N . The generating function of these numbers was studiedby MacMahon [1] and shown to be given, again, by an infinite product: ∞ 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 statistical me-chanics, this generating function becomes the partition function Z = ∞ X N =0 pp( N ) q N = X π q | π | of a canonical ensemble of plane partitions, in which each plane partition π hasan energy proportional to the volume | π | .We shall deform this simplest model by external potentials. To this end, wehave to introduce the notion of “diagonal slices” of a plane partition. Given a plane partition π = ( π ij ) ∞ i,j =1 , the partition π ( m ) = (cid:26) ( π i,i + m ) ∞ i =1 if m ≥ π j − m,j ) ∞ j =1 if m < m -th diagonal slice of π . These partitions { π ( m ) } ∞ m = −∞ representa sequence of 2D Young diagrams that are literally obtained by slicing the 3DYoung diagrams (Figure3). 5igure 3: Diagonal slices (b) of plane partition (a)The diagonal slices are not arbitrary but satisfy the condition [2, 3] · · · ≺ π ( − ≺ π ( − ≺ π (0) ≻ π (1) ≻ π (2) ≻ · · · , (2.3)where “ ≻ ” denotes interlacing relation , namely, λ = ( λ , λ , . . . ) ≻ µ = ( µ , µ , . . . ) def ⇐⇒ λ ≥ µ ≥ λ ≥ µ ≥ · · · . Because of these interlacing relations, a pair (
T, T ′ ) of semi-standard tableauxis obtained on the main diagonal slice λ = π (0) by putting “ m + 1” in boxes ofthe skew diagram π ( ± m ) /π ( ± ( m + 1)).By this mapping π ( T, T ′ ), the partition function Z of the plane par-titions can be converted to a triple sum over the tableau T, T ′ and their shape λ : Z = X λ X T,T ′ :shape λ q T q T ′ , (2.4)where q T = ∞ Y m =0 q ( m +1 / | π ( − m ) /π ( − m − | ,q T ′ = ∞ Y m =0 q ( m +1 / | π ( m ) /π ( m +1) | . By the well known combinatorial definition of the Schur functions [19], thepartial sum over the semi-standard tableaux turn out to be a special value of6he Schur functions: X T :shape λ q T = X T ′ :shape λ q T ′ = s λ ( q ρ ) , (2.5)where q ρ = ( q / , q / , . . . , q n +1 / , · · · ) . Thus the partition function can be eventually rewritten as Z = X λ s λ ( q ρ ) . (2.6)Let us note that the special value of the Schur functions has the so called Hookformula [19] s λ ( q ρ ) = q n ( λ )+ | λ | / Y ( i,j ) ∈ λ (1 − q h ( i,j ) ) − , (2.7)where ( i, j ) stands for the ( i, j )-th box in the Young diagram, and n ( λ ) is givenby n ( λ ) = ∞ X i =1 ( i − λ i . We now deform the foregoing melting crystal model by introducing the externalpotentials Φ k ( λ, p ) = ∞ X i =1 q k ( p + λ i − i +1) − ∞ X i =1 q k ( − i +1) with coupling constants t k , k = 1 , , , . . . , on the main diagonal slice λ = π (0).The right hand side of the definition of Φ k ( λ, p ) is understood to be a finite sum(hence a rational function of q ) by cancellation of terms between the two sums:Φ k ( λ, p ) = ∞ X i =1 ( q k ( p + λ i − i +1) − q k ( p − i +1) ) + q k − q pk − q k . The partition function of the deformed model 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 ) .
7e can repeat the previous calculations in this setting to rewrite the new par-tition function Z p ( t ) as Z p ( t ) = X λ s λ ( q ρ ) q Φ( t,λ,p ) . (2.9)Modifying this partition function slightly, we obtain the instanton sum of5D N = 1 SUSY U (1) Yang-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 ) Q | λ | e Φ( t,λ,p ) . (2.10) q and Q are related to physical parameters R, Λ , ~ of 5D Yang-Mills theory as q = e − R ~ , Q = ( R Λ) . The external potentials represent the contribution of Wilson loops along the fifthdimension [20]. In this sense, Z p ( t ) /Z p (0) is a generating function of correlationfunctions of those Wilson loop operators.Our goal is to show that the partition function Z p ( t ) is, up to a simple factor,the tau function of the (1D) Toda hierarchy. To this end, we now consider afermionic representation of this partition function. Let ψ ( z ) and ψ ∗ ( z ) denote complex 2D fermion fields ψ ( z ) = ∞ X m = −∞ ψ m z − m − , ψ ∗ ( z ) = ∞ X m = −∞ ψ ∗ m z − m . The Fourier modes ψ m and ψ ∗ m of ψ ( z ) and ψ ∗ ( z ) satisfy the anti-commutationrelations { ψ m , ψ ∗ n } = δ m + n, , { ψ m , ψ n } = { ψ ∗ m , ψ ∗ n } = 0 . The Fock space F splits into charge p subspaces F p : F = ∞ M p = −∞ F p . The charge p subspace F p has a unique normalized ground state (charge p vacuum) | p i and an orthonormal basis | λ ; p i labeled by partitions λ . | p i ischaracterized by the vacuum condition ψ m | p i = 0 for m ≥ − p, ψ ∗ m | p i = 0 for m ≥ p + 1 .
8f the partition is of the form λ = ( λ , . . . , λ n , , , . . . ), the associated element | λ ; p i of the basis is obtained from | p i by the action of fermion operators as | λ ; p i = ψ − ( p + λ − − · · · ψ − ( p + λ n − n ) − ψ ∗ ( p − n )+1 · · · ψ ∗ ( p − | p i . They are orthonormal in the sense that their inner products have the normalizedvalues h λ ; p | µ ; q i = δ pq δ λµ . U (1) current and fermionic representation of tau func-tion The U (1) current J ( z ) of the complex fermion system is defined as J ( z ) = : ψ ( z ) ψ ∗ ( z ): = ∞ X k = −∞ J m z − m − , where : : denotes the normal ordering with respect to the vacuum 0 i :: ψ m ψ ∗ n : = ψ m ψ ∗ n − h | ψ m ψ ∗ n | i . The Fourier modes J m = ∞ X n = −∞ : ψ m − n ψ ∗ n :of J ( z ) satisfy the commutation relations[ J m , J n ] = mδ m + n, (3.1)of the A ∞ Heisenberg algebra, and play the role of “Hamiltonians” in the usualfermionic formula of the KP and 2D Toda hierarchies [15, 16]. For the case oftau functions τ ( t, ¯ t ), t = ( t , t , . . . ), ¯ t = (¯ t , ¯ t , . . . ), of the 2D Toda hierarchy,the fermionic 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 dimensional Clifford group GL ( ∞ ). Z p ( t ) The partition function Z p ( t ) of the deformed melting crystal has a fermionicrepresentation of the form Z p ( t ) = h p | G + e H ( t ) G − | p i . (3.3)9et us explain the constituents of this formula along with an outline of thederivation of this formula. H ( t ) is the linear combination H ( t ) = ∞ X k =1 t k H k of the “Hamiltonians” H k = ∞ X n = −∞ q kn : ψ − n ψ ∗ n : . The aforementioned basis elements | λ ; p i of the Fermion Fock space turn out tobe eigenvectors of these Hamiltonians. The eigenvalues are nothing but the thepotential 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 special form G ± = exp (cid:16) ∞ X k =1 q k/ k (1 − q k ) J ± k (cid:17) . Since the numerical factors q k/ / (1 − q k ) in this definition can be expanded as q k/ − q k = − X m = −∞ q − k ( m +1 / = ∞ X m =0 q k ( m +1 / , one can factorize these operators as G + = − Y m = −∞ Γ + ( m ) , G − = ∞ Y m =0 Γ − ( m ) , (3.5)where Γ ± ( m ) = exp (cid:16) ∞ X k =1 k q ∓ k ( m +1 / J ± k (cid:17) . These Γ ± ( m )’s are a specialization of the so called vertex operators V ± ( z ) = exp (cid:16) ∞ X k =1 z k k J ± k (cid:17) for bosonization of the complex fermions.Following the idea of Okounkov and Reshetikhin [2], we now consider m asa fictitious “time” variable. A plane partition then may be thought of as the10path” (or “world volume”) of discrete time evolutions of a partition λ thatstarts from the empty partition ∅ = (0 , , . . . ) at infinite past and ends again in ∅ at infinite future (see Figure3). The vertex operators Γ ± ( m ) play the role oftransfer matrices between neighboring diagonal slices.The vertex operators Γ ± ( m ) act on the aforementioned orthonormal bases | λ ; p i and h λ ; p | as h λ ; p | Γ + ( m ) = X µ ≻ λ h µ ; p | q − ( m +1 / | µ |−| λ | ) (3.6)for m = − , − , . . . andΓ − ( m ) | λ ; p i = X µ ≻ λ q ( m +1 / | µ |−| λ | ) | µ ; p i (3.7)for m = 0 , , . . . [2, 3]. The right hand side of these formulas give a linearcombination of all possible time evolutions of the m -th slice λ = π ( m ) at thenext time. The weight q ∓ ( m +1 / | µ |−| λ | ) of each state on the right hand sideare exactly the factors assigned to the boxes of π ( m ) /π ( m ∓
1) in the definitionof the weights q T , q T ′ that appear in the combinatorial formula (2.4).Since G ± are products of these slice-to-slice “transfer matrices”, h p | G + and G − | p i become linear combinations of the states h λ ; p | and | λ ; p i that evolve fromthe ground states h p | and | p i at m = ∓∞ . By what we have seen above, theweights of h λ ; p | and | λ ; p i in these linear combinations are given by the partialsums of q T and q T ′ over all semi-standard tableaux T and T ′ of shape λ , namely,the special value s λ ( q ρ ) of the Schur function. Thus h p | G + and G − | p i can beexpressed as h p | G + = X λ X T :shape λ q T h λ ; p | = X λ s λ ( q ρ ) h λ ; p | , (3.8) G − | p i = X λ X T ′ :shape λ q T ′ | λ ; p i = X λ s λ ( q ρ ) | λ ; p i . (3.9)The expectation value of e H ( t ) with respect to these states yields the fermionicrepresentation (3.3) of the partition function Z p ( t ).The fermionic representation (3.3) is apparently different from the fermionicformula (3.2) of tau functions of the 2D Toda hierarchy. To show that Z p ( t ) isindeed a tau function, we have to rewrite (3.3) to the form of (3.2). This is theplace where the quantum torus Lie algebra joins the game.11 Quantum torus Lie algebra
Let V ( k ) m ( k = 0 , , . . . , m ∈ Z ) denote the following fermion bilinear forms: V ( k ) m = q − km/ ∞ X n = −∞ q kn : ψ m − n ψ ∗ n := q k/ I dz πi z m : ψ ( q k/ z ) ψ ∗ ( q − k/ z ):Note that J m = V (0) m , H k = V ( k )0 . Actually, V ( k ) m coincides with Okounkov and Pandharipande’s operator E m ( z )[21, 22] specialized to z = q k . As they found for E m ( z ), our V ( k ) m ’s satisfy thecommutation relations[ V ( k ) m , V ( l ) n ] = ( q ( lm − kn ) / − q ( kn − lm ) / )( V ( k + l ) m + n − δ m + n, q k + l − q k + l ) . (4.1)This is a (central extension of) q -deformation of the Poisson algebra of functionson a 2-torus. We refer to this Lie algebra as “quantum torus Lie algebra”. Moreprecisely, a full quantum torus Lie algebra should contain elements for k < The following relations, which we call “shift symmetry”, play a central role inidentifying Z p ( t ) as a tau function: G − G + (cid:16) V ( k ) m − δ m, q k − q k (cid:17) ( G − G + ) − = ( − k (cid:16) V ( k ) m + k − δ m + k, q k − q k (cid:17) (4.2)These relations are derived as follows.Let us recall that the fermion fields ψ ( z ) , ψ ∗ ( z ) transform under adjointaction by J ± k ’s asexp (cid:16) ∞ X k =1 c k J ± k (cid:17) ψ ( z ) exp (cid:16) − ∞ X k =1 c k J ± k (cid:17) = exp (cid:16) ∞ X k =1 c k z ± k (cid:17) ψ ( z ) , (4.3)exp (cid:16) ∞ X k =1 c k J ± k (cid:17) ψ ∗ ( z ) exp (cid:16) − ∞ X k =1 c k J ± k (cid:17) = exp (cid:16) − ∞ X k =1 c k z ± k (cid:17) ψ ∗ ( z ) . (4.4)12y letting c k = q k/ / (1 − q k ), the exponential operators in these formulas turninto G ± , so that we have the operator identities G + ψ ( z ) G + − = ( q / z ; q ) − ∞ ψ ( z ) , (4.5) G + ψ ∗ ( z ) G + − = ( q / z ; q ) ∞ ψ ∗ ( z ) , (4.6) G − ψ ( z ) G −− = ( q / z − ; q ) − ∞ ψ ( z ) , (4.7) G − ψ ∗ ( z ) G −− = ( q / z − ; q ) ∞ ψ ∗ ( z ) , (4.8)where ( z ; q ) ∞ denotes the standard q -factorial symbol( z ; q ) ∞ = ∞ Y n =0 (1 − zq n ) . We use these operator identities to derive transformation of the fermionbilinear forms : ψ ( q k/ z ) ψ ∗ ( q − k/ z ): under conjugation by G ± . Since: ψ ( q k/ z ) ψ ∗ ( q − k/ z ): = − ψ ∗ ( q − k/ z ) ψ ( q k/ z ) + q k/ (1 − q k ) z , let us first consider ψ ∗ ( q − k/ z ) ψ ( q k/ z ). Under conjugation by G + , it trans-forms as G + ψ ∗ ( q − k/ z ) ψ ( q k/ z ) G + − = ( q / · q − k/ z ; q ) ∞ ( q / · q k/ z ; q ) ∞ ψ ∗ ( q − k/ z ) ψ ( q k/ z )= k Y m =1 (1 − zq ( k +1) / − m ) ψ ∗ ( q − k/ z ) ψ ( q k/ z ) . This implies that G + (cid:16) : ψ ( q k/ z ) ψ ∗ ( q − k/ z ): − q k/ (1 − q k ) z (cid:17) G + − = k Y m =1 (1 − q ( k +1) / − m z ) (cid:16) : ψ ( q k/ z ) ψ ∗ ( q − k/ z ): − q k/ (1 − q k ) z (cid:17) . (4.9)In much the same way, we can derive a similar transformation under conjugationby G − . In this case, it is more convenient to rewrite the result as follows: G −− (cid:16) : ψ ( q k/ z ) ψ ∗ ( q − k/ z ): − q k/ (1 − q k ) z (cid:17) G − = k Y m =1 (1 − q − ( k +1) / m z − ) (cid:16) : ψ ( q k/ z ) ψ ∗ ( q − k/ z ): − q k/ (1 − q k ) z (cid:17) . (4.10)13e note here that the prefactors on the right hand side of the last twoequations are related as k Y m =1 (1 − q ( k +1) / − m z ) = ( − z ) k k Y m =1 (1 − q − ( k +1) / m z − ) . Accounting for this simple, but significant relation, we can derive the identity G − G + (cid:16) : ψ ( q k/ z ) ψ ∗ ( q − k/ z ): − q k/ (1 − q k ) z (cid:17) ( G + G − ) − = ( − z ) k (cid:16) : ψ ( q k/ z ) ψ ∗ ( q − k/ z ): − q k/ (1 − q k ) z (cid:17) . (4.11)The shift symmetry (4.2) follows immediately from this identity.When m = 0 and m = − k , (4.2) takes the particular form G − G + (cid:16) V ( k )0 − q k − q k (cid:17) ( G − G + ) − = ( − k V ( k ) k , (4.12)( G − G + ) − (cid:16) V ( k )0 − q k − q k (cid:17) G − G + = ( − k V ( k ) − k . (4.13)It is these identities that we shall use to convert the fermionic representation of Z p ( t ) to the standard fermionic formula of tau functions. Let us split the operator G + e H ( t ) G − in (3.3) into three pieces as G + e H ( t ) G − = G + e H ( t ) / e H ( t ) / G − = G + e H ( t ) / G + − · G + G − · G −− e H ( t ) / G − and use the special cases (4.12) and (4.13) of the shift symmetry to rewrite thosepieces.To this end, it is convenient to rewrite (4.12) and (4.13) as G + (cid:16) H k − q k − q k (cid:17) G + − = ( − k G −− V ( k ) k G − ,G −− (cid:16) H k − q k − q k (cid:17) G − = ( − k G + V ( k ) − k G + − . Though the operators V ( k ) ± k on the right hand side are unfamiliar in the theoryof integrable hierarchies, we can convert them to the familiar “Hamiltonians” J ± k = V (0) ± k of the Toda hierarchies as q W/ V ( k ) k q − W/ = V (0) k = J k , q − W/ V ( k ) − k q W/ = V (0) − k = J − k , (5.1)14here W is a special element of W ∞ algebra: W = W (3)0 = ∞ X n = −∞ n : ψ − n ψ ∗ n :We thus eventually obtain the relations G + (cid:16) H k − q k − q k (cid:17) G + − = ( − k G −− q − W/ J k q W/ G − , (5.2) G −− (cid:16) H k − q k − q k (cid:17) G − = ( − k G + q W/ J − k q − W/ G + − (5.3)between H k ’s and J ± k ’s.By these relations, G + e H ( t ) / G + − can be calculated as G + e H ( t ) / G + − = exp (cid:16) ∞ X k =1 t k q k − q k ) (cid:17) G −− q − W/ exp (cid:16) ∞ X k =1 ( − k t k J k (cid:17) q W/ G − . A similar expression can be derived for G −− e H ( t ) G − as well. We can thusrewrite G + e H ( t ) G − as G + e H ( t ) G − = exp (cid:16) ∞ X k =1 t k q k − q k (cid:17) G −− q − W/ exp (cid:16) ∞ X k =1 ( − k t k J k (cid:17) ×× g exp (cid:16) ∞ X k =1 ( − k t k J − k (cid:17) q − W/ G + − where g = q W/ ( G − G + ) q W/ . (5.4)The partition function Z p ( t ) is given by the expectation value of this operatorwith respect to h p | and | p i . Since the action by the leftmost and rightmost piecesof g yields only a scalar multiplier to h p | , | p i as h p | G −− q − W/ = q − p ( p +1)(2 p +1) / h p | , (5.5) q − W/ G + − | p i = q − p ( p +1)(2 p +1) / | p i , (5.6) Z p ( t ) can be expressed as Z p ( t ) = exp (cid:16) ∞ X k =1 t k q k − q k (cid:17) q − p ( p +1)(2 p +1) / ×× h p | exp (cid:16) ∞ X k =1 ( − k t k J k (cid:17) g exp (cid:16) ∞ X k =1 ( − k t k J − k (cid:17) | p i . (5.7)The expectation value h p | · · · | p i takes exactly the form of (3.2). Thus, up to thesimple prefactor, Z p ( t ) is essentially a tau function of the 2D Toda hierarchy.Thus we find that an integrable structure behind the melting crystal model isthe 2D Toda hierarchy. This is, however, not the end of the story.15 .2 1D Toda hierarchy as true integrable structure The foregoing calculation is based on the splitting G + e H ( t ) G − = G + e H ( t ) / G + − · G + G − · G −− e H ( t ) / 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 + − · G + G − = G + G − · G −− e H ( t ) G − This leads to another set of expressions of Z p ( t ) in which only the h p | · · · | p i partis different. We thus have the following three different expressions for this part: h p | exp (cid:16) ∞ X k =1 ( − k t k J k (cid:17) g exp (cid:16) ∞ X k =1 ( − k t k J − k (cid:17) | p i = h p | exp (cid:16) ∞ X k =1 ( − k t k J k (cid:17) g | p i = h p | g exp (cid:16) ∞ X k =1 ( − k t k J − k (cid:17) | p i . (5.8)These identities of the expectation values can be directly derived from theoperator identities J k g = gJ − k , k = 1 , , , . . . (5.9)satisfied by g . (These operator identities themselves are a consequences of theshift symmetry of V ( k ) m ’s.) Generally speaking, this kind of operator identitiesimply symmetry constrains on the tau functions [23, 24]; in the present case,the constraints read ∂∂t k τ p ( t, ¯ t ) + ∂∂ ¯ t k τ p ( t, ¯ t ) = 0 , k = 1 , , , . . . . (5.10)In other words, the tau 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 Toda hierarchy that has a singleseries of time variables t = ( t , t , . . . ) rather than the two series of the 2D Todahierarchy. Thus the 1D Toda hierarchy eventually turns out to be an underlyingintegrable structure of the deformed melting crystal model.The same conclusion can be derived for the instanton sum (2.10) of 5D SUSY U (1) Yang-Mills theory. It has a fermionic representation of the form Z p ( t ) = h p | G + Q L e H ( t ) G − | p i (5.11)where L is a special element of the Virasoro algebra: L = ∞ X n = −∞ n : ψ − n ψ ∗ n : . Z p ( t ) to the form of (5.7). The counterpart of g is given by g = q W/ G − G + Q L G − G + q W/ , (5.12)which, too, satisfy the reduction conditions (5.9) to the 1D Toda hierarchy.Thus a relevant integrable structure is again the 1D Toda hierarchy. In deriving the instanton sum (2.10), 5D space-time is partially compactified inthe fifth dimension as R × S . The parameter R is the radius of S . Therefore,letting R → R →
0. Any naive prescription letting R → t dependence disappears or becomes trivial [12]. Secondly, theshift symmetry of the quantum torus Lie algebra ceases to exist in the limit as q = e − R ~ →
1. Speaking more precisely, the quantum torus Lie algebra itselfturns into a W ∞ algebra in this limit, but no analogue of shift symmetry (4.2)is known for the latter case. For these reasons, the 4D case has to be studiedindependently.The 4D instanton sum [7, 8], too, is a sum over partitions. Moreover, thisstatistical sum has a fermionic representation [14, 9]. Marshakov and Nekrasov[17, 18] further introduced external potentials therein. Actually, the 4D instan-ton sum for U (1) gauge theory is almost identical to the generating function ofGromov-Witten invariants of CP [21, 22]. This can be most clearly seen in thefermionic representation of these generating functions, which reads Z p ( t ) = h p | e J / ~ exp (cid:16) ∞ X k =1 t k P k +1 k + 1 (cid:17) e J − / ~ | p i , (6.1)where P k ’s are fermion bilinear forms introduced by Okounkov and Pandhari-pande for a fermionic representation of (absolute) Gromov-Witten invariantsof CP [21]. As regards these Gromov-Witten invariants (in other words, cor-relation functions of the topological σ model) , it has been known for years[25, 26, 27, 28, 29] that a relevant integrable structure is the 1D Toda hierarchy.Thus the 1D Toda hierarchy is expected to be the integrable structure ofthe 4D instanton sum as well. This has been confirmed by Marshakov andNekrasov in detail [17, 18]. What is still missing, however, is a formula like(5.7) that directly connects Z p ( t ) with the standard fermionic formula (3.2)of the tau function. Finding a 4D analogue of (5.7) is thus an intriguing openproblem. This issue is also closely related to the fate of shift symmetry (4.2) inthe q → .2 Relation to topological strings Our results are directly or indirectly connected with some aspects of topologicalstrings as well.1. According to the theory of topological vertex [5], the partition function Z p ( t ) of the deformed melting crystal model has another interpretation as the A -model topological string amplitude for the toric Calabi-Yau threefold O ⊕O ( − → CP . In this interpretation, q and Q are parametrized by the stringcoupling constant g st and the K¨ahler volume a of CP as q = e − g st , Q = e − a . Specializing the value of t leads to several interesting observations [12].2. A generating function of the two-legged topological vertex W λµ ∼ c λµ • isknown to give a tau function of the 2D Toda hierarchy [30]. In the fermionicrepresentation (3.2), this amounts to the case where g = q W/ G + G − q W/ . (6.2)(Actually, for complete agreement with the usual convention, we have to replace W with K = ∞ X n = −∞ (cid:16) n − (cid:17) : ψ − n ψ ∗ n : , but this is not a serious problem. The difference can be absorbed by rescaling t k ’s.) Let us stress that this GL ( ∞ ) element does not satisfy the reductioncondition (5.9) to the 1D Toda hierarchy.3. A generating function of double Hurwitz numbers for coverings of CP gives yet another type of tau function of the 2D Toda hierarchy [31]. Actually,the GL ( ∞ ) element for the fermionic representation is given by g = q W/ . (6.3)(More precisely, as in the previous case, W has to be replaced by K , but thedifference is again irrelevant.) In this case, the reduction condition (5.9) to the1D Toda hierarchy is not satisfied, but the operator identities (5.1) imply thatanother set of reduction conditions are hidden behind (see below). As a consequence of the shift symmetry of V ( k ) m ’s, the GL ( ∞ ) elements g of theaforementioned models of topological strings turn out to satisfy some algebraicrelations other than (5.9). According to general results on constraints of the 2DToda hierarchy [23, 24], such relations imply the existence of constraints on thetau functions and the Lax and Orlov-Schulman operators. Those constraintsinherit the structure of the quantum torus Lie algebra. Let us illustrate thisobservation for the case of double Hurwitz numbers over CP .18he GL ( ∞ ) element g = q W/ for this case satisfies the operator identities J k g = gV ( k ) k , gJ − k = V ( k ) − k g (6.4)as a consequence of (5.1). These identities can be converted to the constraints L = q / q ¯ M ¯ L, ¯ L − = q − / q M L − (6.5)on the Lax and Orlov-Schulman operators L, M, ¯ L, ¯ M of the 2D Toda hierarchy.Emergence of the exponential operators q M and q ¯ M is a manifestation of thequantum torus Lie algebra. To see this, let us recall that the Lax and Orlov-Schulman operators satisfy the (twisted) canonical commutation relations[ L, M ] = L, [ ¯ L, ¯ M ] = ¯ L. (6.6)This implies that the monomials q − km/ L m q kM and q − km/ ¯ L m q k ¯ M of L, q M , ¯ L, q ¯ M give two copies of realizations of the quantum torus Lie algebra.The constraints (6.5) are remarkably similar to the “string equations” L = ¯ M ¯ L, ¯ L − = M L − (6.7)of c = 1 strings at self-dual radius [32, 33, 23, 24]. A relevant algebraic structureof these string equations is the W ∞ algebra. Thus (6.5) may be thought of as q -deformations of these W -algebraic constraints. Acknowledgements
We are grateful to Nikita Nekrasov and Motohico Mulase for valuable commentsand fruitful discussion. K.T is partly supported by Grant-in-Aid for ScientificResearch No. 18340061 and No. 19540179 from the Japan Society for thePromotion of Science.
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