Intertwining operator and integrable hierarchies from topological strings
aa r X i v : . [ h e p - t h ] J a n Intertwining operator and integrable hierarchiesfrom topological strings
Jean-Emile Bourgine
Korea Institute for Advanced Studies (KIAS)Quantum Universe Center (QUC)85 Hoegiro, Dongdaemun-gu, Seoul, South Korea [email protected]
Abstract
In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topo-logical strings vertex provides a tau-function of the KP hierarchy after an appropriate timedeformation. We revisit their derivation with a focus on the underlying quantum W ∞ sym-metry. Specifically, we point out the role played by automorphisms and the connection withthe intertwiner - or vertex operator - of the algebra. This algebraic perspective allows usto extend part of their derivation to the refined melting crystal model, lifting the algebra tothe quantum toroidal algebra of gl (1) (also called Ding-Iohara-Miki algebra). In this way, wetake a first step toward the definition of deformed hierarchies associated to A-model refinedtopological strings. Introduction
The infinite Lie algebra b gl ( ∞ ) and its fermionic representation plays a key role in the constructionof solutions for the Kadomtsev-Petviashvili (KP) hierarchy [2]. This role was emphasized by theKyoto school [3], and led to the introduction of several reductions of the KP hierarchy associated tosubalgebras of b gl ( ∞ ). This algebraic approach was further formalized by Kac and Wakimoto, andextended to associate a hierarchy of differential equations to any Kac-Moody algebra [4, 5]. Theseworks illustrate the efficiency of the algebraic approach for integrable hierarchies. Another indisputable success of the algebraic approach is in the study of 2D Conformal FieldTheories (CFT) where most physical quantities are determined by the symmetry algebras [8]. Thisformidable success led Frenkel and Reshetikhin to propose an extension of the vertex operatortechnique employed in 2D CFT to some quantum integrable systems using their quantum affinealgebra of symmetries. Indeed, algebraically, a vertex operator is simply an intertwiner betweena Fock (or level one) representation F of the algebra and its tensor product V ⊗ F with a levelzero representation V . For the original vertex operator of a free boson, the algebra is a Heisenbergalgebra, it becomes an affine Lie algebra in Wess-Zumino-Witten models. In the case of quantumaffine algebras, the Frenkel-Jing [9] construction provides the level one representation while the levelzero representation is the usual highest weight representation acting on the quantum spins. Thisvertex operator technique has been applied, for instance, to the diagonalization of the infinite XXZspin chain Hamiltonian in [10]. To avoid the confusion with exponentials of free fields, the vertexoperator is often called intertwining operator , or simply intertwiner , in this context.In [11], Awata, Feigin and Shiraishi (AFS) applied the vertex operator technique for a differentalgebra called quantum toroidal gl (1) (or Ding-Iohara-Miki) algebra [12, 13]. They have shownthat, quite remarkably, the matrix elements of the intertwiner reproduce the refined vertex of thetopological string theory [14–16]. This observation led to a number of important results in this field.For example, we can mention the extension of the topological vertex technique to various theories[17–24] and observables [25–27], the derivation of proofs for the q-deformed AGT correspondence[28–30], or the description of the fiber-base duality [30–33].In this paper, we investigate the role played by quantum algebras in the well-known relationbetween self-dual topological strings and integrable hierarchies [34]. Refined topological stringsdepend on two parameters ( q, t − ), they are identified with the parameters ( q , q ) of the quantumtoroidal gl (1) algebra. In the self-dual limit t → q (or q q → q is identified with the exponentiated string couplingconstant e g str and the algebra reduces to the quantum W ∞ algebra [35,36]. The latter is equivalentto the b gl ( ∞ ) algebra mentioned earlier upon a linear transformation of the generators [37–39], it isthus naturally expected to be involved in the relation with integrable hierarchies, which was indeed For a brief introduction, see the excellent reviews [6, 7] or the reference book [5]. W ∞ algebra in this setup has already been describedin [1], we revisit here their derivation with an extra emphasis on certain algebraic structures.Specifically, we underline the role of two particular objects, the first one being the AFS intertwiningoperator introduced previously, and the second one is an operator associated to the framing factorsof the topological vertex that we call framing operator for short. Algebraically, it realizes the actionof a certain automorphism T in the Fock module of the algebra. As we shall see, these two objectsare deeply related to the SL (2 , Z ) subgroup of automorphisms. Both are essential ingredientsunderlying the Nakatsu-Takasaki derivation.The main motivation for our study is the search for a deformation of integrable hierarchies thatwould correspond to the refinement of topological strings in their A-model formulation. Naively,it would seem difficult as the fermionic structure disappears from the Fock representation, andonly a bosonic formulation subsists. Yet, the presence of the quantum toroidal algebra hints forthe preservation of an integrable structure. It brings us to the second part of our paper in whichwe attempt to refine the Nakatsu-Takasaki derivation. The definition and properties of the twoprevious algebraic objects are easily extended to the quantum toroidal case, and we manage towrite the generating function in the canonical form h∅| e − P k k τ k J k g |∅i , (1.1)where J k are the modes of a Heisenberg algebra, |∅i the vacuum of the Fock space and τ k the(rescaled) times of the hierarchy. Unfortunately, we were unable to show that g is a group-likeelement of GL ( ∞ ), that would imply that the generating function is a tau function of an integrablehierarchy. And, in fact, it cannot be group-like as a perturbative analysis indicates that the Hirotaequation is no longer satisfied in the refined case. Nevertheless, we are confident that this result isbringing us closer to the definition of refined hierarchies.The organization of the paper is very straightforward: the first part deals with self-dual topo-logical strings and the quantum W ∞ algebra while the second part presents the refined casecorresponding to the quantum toroidal algebra. The appendix contains the proofs for several iden-tities used in the main text. 2 Integrable structure of the melting crystal model W ∞ algebra The quantum W ∞ algebra can be presented in many ways and found under different names, e.g.quantum torus algebra in [1], trigonometric Sin-Lie algebra in [37,41],... Here, we use a presentationin terms of generators W m,n with integer indices, and two central elements ( c , c ), that satisfy thecommutation relations[ W m,n , W m ′ ,n ′ ] = ( q m ′ n − q mn ′ ) (cid:18) W m + m ′ ,n + n ′ + c δ m + m ′ − q n + n ′ − c δ n + n ′ − q m + m ′ (cid:19) . (2.1)In fact, this algebra is a central extension by the element c of the quantum algebra used in [1] (upto a rescaling of the generators). In application to integrable hierarchies, we consider mostly therepresentation of levels (1 , p, q )-brane web construction [44, 45].The algebra has the group of automorphisms GL (2 , Z ) [38], but we will be only interested inthe SL (2 , Z ) subgroup generated by the transformations S and T . These transformations act asfollows on the generators, S : W m,n → q − ( m +1) n W − n,m , ( c , c ) → ( c , − c ) , T : W m,n → q − n / W m − n,n + c δ m − n − q n / δ m, − q n (1 − δ n, ) − c δ n, − q m (1 − δ m, ) ! , ( c , c ) → ( c , − c + c ) . (2.2) The quantum W ∞ algebra has a representation of levels (1 ,
0) on the Fock space F of a Diracfermion. We refer the reader to [39] for a recent review of this representation. The Fock space isbuilt from a vacuum state |∅i annihilated by the positive modes of the fermionic fields ψ ( z ) = X r ∈ Z +1 / z − r − / ψ r , ¯ ψ ( z ) = X r ∈ Z +1 / z − r − / ¯ ψ r , { ψ r , ¯ ψ s } = δ r + s . (2.3) There are different conventions for the fermionic modes and we use here half-integer indices. To recover theconvention employed in [1], one should simply replace ψ r → ψ ∗ r +1 / and ¯ ψ r → ψ r − / . ψ ( z ) =: e φ ( z ) :, ψ ( z ) =: e − φ ( z ) :, and : ¯ ψ ( z ) ψ ( z ) := ∂φ ( z ) involvingthe bosonic field φ ( z ) = Q + J log z − X k ∈ Z × k z − k J k , [ J k , J l ] = kδ k + l , [ Q, J ] = 1 , (2.4)the Fock space F has an alternative construction obtained by the action of the negative modes J − k on the vacuum state |∅i . The vacuum is annihilated by positive modes J k , and the normal orderingnaturally consists in moving these modes to the right. The Fock space F can be decomposedaccording to the values of the zero mode J into F = L k ∈ Z F k . Since the representation of thegenerators W m,n is neutral, we can restrict ourselves to the subspace F of zero charge. Becauseof this restriction, we will only be able to discuss the KP hierarchy (not mKP), but it makes thequestion of refinement simpler. Symmetric polynomials
The vector space F is isomorphic to the space of symmetric polynomi-als with infinitely many variables. The isomorphism sends the vacuum |∅i to the trivial polynomial1, and maps the PBW basis to products of elementary powers sums p k ( x ) = P k x ki ,( J − λ ) k · · · ( J − λ n ) k n |∅i ↔ p λ ( x ) k · · · p λ n ( x ) k n . (2.5)Thus, the negative modes J − k act as a multiplication by p k while positive modes J k act as k∂/∂p k on symmetric polynomials. This isomorphism is useful in order to define the Schur basis of states | λ i , labeled by a partition λ , and obtained as the inverse image of the Schur polynomials s λ ( x ) (theyform a basis of the ring of symmetric polynomials). These Schur states coincide with the fermionicPBW basis of F . In the same way, we define the Macdonald basis with states | P λ i obtained asthe inverse image of the Macdonald polynomials P λ ( x ). We also define the dual states h λ | and h P λ | with the scalar product h λ || µ i = δ λ,µ , h P λ || P µ i = h P λ , P λ i q,t δ λ,µ , (2.6)where h P λ , P λ i q,t is the norm square of Macdonald scalar product [46]. Explicitly, h P λ , P λ i q,t = Y ∈ λ − q a ( )+1 t l ( ) − q a ( ) t l ( )+1 , (2.7)where a ( ) = λ i − j and l ( ) = λ ′ j − i are the arm and leg length of the box ( i, j ) in the Youngdiagram representing the partition λ = ( λ , λ , · · · ). We denote λ ′ the partition corresponding tothe transposed Young diagram of λ . Action of the algebra
We denote ρ ( D ) u,v the representation of the quantum W ∞ algebra oflevels (1 ,
0) and weights u, v ∈ C × on the Fock space F . To get rid of the weights dependence, we4ometimes use the notation ¯ W m,n = ρ ( D )1 , ( W m,n ) so that the general representation reads ρ ( D ) u,v ( W m,n ) = u n v m ¯ W m,n − − u n − q n δ m, (1 − δ n, ) . (2.8)The action on F can be expressed in different ways, the simplest one is in terms of the fermionicfields, ¯ W m,n = X r ∈ Z +1 / q − ( r +1 / n : ¯ ψ m − r ψ r : . (2.9)Instead, in the bosonic presentation the modes J k are identified with the generators ¯ W k, whilethe other generators define currents represented as vertex operators. Fortunately, we will not needthese expressions here. Finally, the action of the generators W m,n on the Schur basis can be writtenexplicitly. When m = 0, these operators add or remove strips of | m | boxes to the Young diagramlabeling the states [39]. On the other hand, the action of the modes W ,n is diagonal and read ρ ( D ) u,v ( W , − k ) | λ i = φ k ( λ, s ) | λ i , φ k ( λ, s ) = − (1 − q k ) q ks X ( i,j ) ∈ λ q − ( i − j ) k − − q ks − q − k . (2.10)The r.h.s. is independent of the weight v , but depends on the weight u = q − s through the variable s . The eigenvalues φ k ( λ, s ) are coupled to the time evolution of the hierarchy in [1, 47]. Framing operator
In addition to the generators ¯ W m,n , it is necessary to introduce the followingoperators acting on the Fock space F , J = X r ∈ Z +1 / : ¯ ψ − r ψ r : , L = X r ∈ Z +1 / r : ¯ ψ − r ψ r : , W = X r ∈ Z +1 / r : ¯ ψ − r ψ r : . (2.12)These operators act diagonally on the Schur states, with the eigenvalues given by J | λ i = 0 , L | λ i = | λ | | λ i , W | λ i = κ ( λ ) | λ i . (2.13)Here | λ | denotes the number of boxes in the Young diagram of λ and κ ( λ ) = 2 P ( i,j ) ∈ λ ( j − i ).As we shall see, the insertion of the operator Q L inside a bosonic correlator is interpreted as the These operators can be obtained using the expansion at q = e ε → W ,n = X r ∈ Z +1 / q − n ( r +1 / : ¯ ψ − r ψ r := J − nε (cid:18) L + 12 J (cid:19) + 12 n ε (cid:18) W + L + 14 J (cid:19) + O ( ε ) . (2.11)They correspond to the zero-modes of the currents of spin one, two (a.k.a. Virasoro) and three of the W ∞ algebrathat appears in the degenerate limit q →
1. Note that there is a small mismatch with respect to the definitions givenin [1, 47, 48] as we introduced shifts by zero modes of lower spin for later convenience. Our operator W coincideswith the cut-and-join operator of ref. [49]. Q . In the same way, the insertion of the diagonal operator q − nW / introduces a framing factor q − nκ ( λ ) / . This factor is related to the Chern-Simons factor atlevel n in the dual 5D N = 1 gauge theory, Z CS ( n ) = Y ( i,j ) ∈ λ v n q ( i − j ) n = v n | λ | q − nκ ( λ ) / . (2.14)The extra dependence in the weight v can be removed using the operator v − nL , it is usuallyabsorbed in the definition of the instanton counting parameter [27].It has been observed in [27] that framing factors follow from the action of the automorphism T in the algebraic framework. Thus, we expect the operator W to be somehow associated tothis automorphism. To formulate this relation, it is more convenient to introduce the combination˜ W = ( W + L ) /
2, and we will call framing operators the operators of the form q α ˜ W for α ∈ R .The shift by L is related to the presence of the parameter v in the Chern-Simons factor 2.14 thatwill be set to v = q − / later on. Then, using the free fermion realization, one can show that q α ˜ W ¯ W m,n q − α ˜ W = q αm / ¯ W m,n + αm . (2.15)When α is an integer, the r.h.s. coincides with the representation of the generators ST α S − · W m,n .In this case, the adjoint action of q α ˜ W realizes the S -dual action of the automorphism T α on theFock space F . Very remarkably, this formula also makes sense for α ∈ R , somehow extending theautomorphism to non-integer values. The relation 2.15 has been derived in [1, 47, 48] and called a shift symmetry , but the connection with the action of the automorphisms appears to be new. The AFS intertwiner has been introduced directly in the context of the refined topological vertex[11]. However, it is relatively easy to perform the self-dual limit t → q of this object and obtain theformulation relevant to the unrefined case [33]. Since the quantum toroidal gl (1) algebra reducesto quantum W ∞ in this limit, we obtain an intertwiner between modules of the latter. Theautomorphisms S and T are then used map these modules to the Fock space F , twisting the Diracrepresentation in the process. As a result, the intertwiner Φ is defined as the operator F ⊗ F → F ρ ( D ) u ,v ( T · W )Φ( v ) = Φ( v ) (cid:0) ρ ( D ) u ,v ◦ S ⊗ ρ ( D ) u ,v ∆( W ) (cid:1) , (2.16)where W denotes any element of the quantum W ∞ algebra. The co-algebraic structure of thequantum toroidal algebra trivializes in the self-dual limit and thus the coproduct in the r.h.s. is theco-commutative one, i.e. ∆( W ) = W ⊗ ⊗ W . The intertwiner depends on a free parameter v , andthe weights of the representation are required to obey the three constraints v = v , u = qvv ,and u = − q / vv u . It is worth mentioning that a dual intertwiner Φ ∗ : F → F ⊗ F is alsointroduced in [11] even though it will not be needed here.The first space F in the tensor product F ⊗ F corresponds to the vertical module in thelanguage of the toroidal algebra, it is associated to the preferred direction of the topological vertexand it plays the role of the level zero representation for the vertex operator. Even in the self-duallimit, the formalism retains this notion of a preferred direction, and it is useful to introduce anotation to distinguish it. Following our earlier works [26, 27, 32], we denote with a double ket(e.g. | λ ii ) the vectors of this module. The solution of the AFS equation 2.16 is nicely expressed bydecomposition on the Schur basis of the vertical module, each component corresponding to a vertexoperator Φ λ : F → F ,Φ( v ) = X λ hh λ | ⊗ Φ λ ( v ) , Φ λ ( v ) = t λ ( v ) : Φ ∅ ( v ) Y ( i,j ) ∈ λ η ( vq i − j ) : . (2.17)In the second equation, the component Φ λ ( v ) has been decomposed into a normalization factor t λ ,a vacuum contribution and a dressing by vertex operators η ( z ),Φ ∅ ( v ) =: exp X k ∈ Z × v − k k (1 − q k ) J k ! : , η ( z ) =: exp − X k ∈ Z × z − k k (1 − q − k ) J k ! : . (2.18)The normalization factor simplifies if we impose the extra constraint q − / u v = v v / amongthe weights: it no longer depends on v and simply writes t λ = s λ ′ ( q − ρ ) with ρ = ( − / , − / , · · · ). It may be useful to make a short historical comment about this equation. In 2D CFT, primary operators ofconformal dimension h satisfy the equation [ L n , φ h ( z )] = ( z n +1 ∂ z + h ( n + 1) z n ) φ h ( z ) with n ≥ L n . This equation can be written in the form 2.16, i.e. ( ρ ⊗ ρ ∆( L n )) φ h ( z ) = φ h ( z ) ρ ( L n ), where ρ is a Fock representation and ρ ( L n ) = − z n +1 ∂ z − h ( n + 1) z n is the level zero representationthat describes the action of the conformal symmetry on the coordinates (the coproduct is also co-commutative).In addition, vertex operators of the free boson obey a similar equation with the Heisenberg algebra [ J n , V α ( z )] = − αz n V α ( z ) where the r.h.s. is trivially a representation of level zero. This equation is generalized further to Wess-Zumino-Witten models and affine Lie algebras [50]. The notation s λ ′ ( q − ρ ) refers to the evaluation of the Schur polynomial s λ ′ ( x ) for the variables ( x , x , · · · ) =( q / , q / , ... ). We assume | q | < hift symmetries The key ingredients in the derivation of the tau function from the meltingcrystal are a set of relations called shift symmetries in [1, 47, 48]. We have already encountered oneof these relations in 2.15. The other relations are of a different nature: they follow from the AFSintertwining equation 2.16 by projection on the vacuum of the vertical module. In order to showthis fact, we introduce the vertical decomposition 2.17 inside the equation 2.16 and project it onthe component λ , ρ ( D ) u ,v ( T · W )Φ λ ( v ) = Φ λ ( v ) ρ ( D ) u ,v ( W ) + X µ Φ µ ( v ) hh µ | ρ ( D ) u ,v ( S · W ) | λ ii . (2.19)Then, we observe that the action of the modes W m,n with m > | λ i is toremove strips of m boxes to the partition λ , and, in particular, these modes annihilate the vacuum |∅ii [39]. Applying this property to the S -dual modes S · W m,n = q − ( m +1) n W − n,m in the previousequation, we obtain an exchange relation for the vacuum component λ = ∅ , ρ ( D ) u ,v ( T · W m,n )Φ ∅ ( v ) = Φ ∅ ( v ) ρ ( D ) u ,v ( W m,n ) , n < . (2.20)This exchange relation reproduces the two missing shift symmetries, as will become clear once wereview the connection between the intertwiner and the melting crystal formalism. Melting crystal
In [40], Okounkov, Reshetikhin and Vafa (ORV) discovered an intriguing con-nection between the topological vertex [14] and plane partitions. They interpreted the vertex asthe generating function of plane partitions with fixed asymptotics given by the three Young dia-grams λ, µ, ν labeling the vertex C λ,µ,ν . As a result, up to a normalization factor, the topologicalvertex counts the configurations of an infinite cube with boxes removed at the corner, effectivelydescribing a melting crystal. This analogy with the melting crystal follows from the rewriting ofthe topological vertex as a correlator of operators acting in the free boson Fock space F . In orderto point out the connection with the intertwiner Φ( v ), we briefly sketch their derivation. It startsfrom the well-known formula of the topological vertex written in terms of skew-Schur polynomials C λ,µ,ν = q − κ ( λ ) / − κ ( ν ) / s ν ′ ( q − ρ ) X η s λ ′ /η ( q − ρ − ν ) s µ/η ( q − ρ − ν ′ ) . (2.21)The argument q − ρ − ν of the skew-Schur functions indicates the evaluation of the polynomial at( x , x , · · · ) = ( q − ν +1 / , q − ν +3 / , · · · ). The rewriting of the topological vertex is based on therealization of skew-Schur polynomials as the matrix elements of the operators Γ ± ( x ) in the Schurbasis, s λ ′ /η ( x ) = h λ ′ | Γ − ( x ) | η i , s µ/η ( x ) = h η | Γ + ( x ) | µ i , Γ ± ( x ) = exp X k> k p k ( x ) J ± k ! . (2.22)8e refer the reader to the appendix of ref. [39] for a short derivation of these well-known formulas.Then, the sum over partitions η in the expression 2.21 of the topological vertex can be performedusing the closure relation of the Schur basis, leading to C λ,µ,ν = q − κ ( λ ) / − κ ( ν ) / s ν ′ ( q − ρ ) h λ ′ | Γ − ( q − ν ′ − ρ )Γ + ( q − ν − ρ ) | µ i . (2.23)In order to build the 3d partitions of the melting crystal, the two operators Γ ± ( x ) need to beexchanged inside the correlator, it produces an extra factor that is easily computed from theirbosonic expression, C λ,µ,ν = ( − | ν | q − κ ( λ ) / Z ( q ) − ( s ν ′ ( q − ρ )) − h λ ′ | Γ + ( q − ν ′ − ρ )Γ − ( q − ν − ρ ) | µ i , (2.24)where Z ( q ) is MacMahon’s generating function of plane partitions, Z ( q ) = X π ∈ P.P. q | π | = ∞ Y n =1 (1 − q n ) − n . (2.25)From its bosonic expression 2.18, the intertwiner Φ( v ) at v = q − / is identified with the operatorinside the correlators after normal-ordering,Φ ν ( q − / ) = t ν Γ − ( q − ν − ρ )Γ + ( q − ν ′ − ρ ) ⇒ C λ,µ,ν = q − κ ( λ ) / − κ ( ν ) / h λ ′ | Φ ν ( q − / ) | µ i . (2.26)Once we set v = q − / , taking into account the various constraints, only two weights are free tochoose in the intertwining relation 2.16, e.g. u and v , and the others are fully determined: u = − u /v , v = v , u = q / v , v = q / v /u . In the following, we denote forsimplicity Φ ν = Φ ν ( q − / ). In fact, other values of the parameter v can be considered by insertionof the operator Q L that acts on the modes as Q L J k Q − L = Q − k J k and so Q L Φ λ ( v ) Q − L = Φ λ ( Qv ).Coming back to the vacuum component, we have found that Φ ∅ = Γ − ( q − ρ )Γ + ( q − ρ ) and thus theexchange relation 2.20 does indeed coincide with the shift symmetries derived by Nakatsu andTakasaki. The starting point for the construction of the KP tau function is MacMahon’s generating functionthat can be written as a sum of Schur polynomials using the Cauchy identity. This expression isthen deformed by the introduction of the time parameters t = ( t k ) coupled to the eigenvalues 2.10of the operator W , − k , Z ( q ) = X λ ( s λ ( q − ρ )) → Z ( q, s, t ) = X λ ( s λ ( q − ρ )) Q | λ | e P k> t k φ k ( λ,s ) . (2.27) For convenience, the overall factor Q s ( s +1) has been removed with respect to the formula (4.4) given in [47].
9e do not indicate the Q -dependence as we can think of it as an extra time parameter t = log Q .To show that the quantity on the right is a tau function of the KP hierarchy, we need to rewrite itas a bosonic correlator in the canonical form 1.1.The first step is similar to the rewriting of the topological vertex in the melting crystal picture,namely the Schur functions are replaced by bosonic correlators using the formulas 2.22, and thetime dependence follows from the diagonal action of W , − k and L on the Schur states, Z ( q, s, t ) = X λ h∅| Γ + ( q − ρ ) ρ ( D ) q − s ,v ( e H ( t ) ) | λ i h λ | Q L Γ − ( q − ρ ) |∅i , H ( t ) = X k> t k W , − k . (2.28)From our previous remark, it is clear that the function Z ( q, s, t ) does not depend on the weight v and we take v = 1. The sum over partitions λ is eliminated using the closure relation of the Schurbasis in order to write Z ( q, s, t ) = h∅| Γ + ( q − ρ ) ρ ( D ) q − s , ( e H ( t ) ) Q L Γ − ( q − ρ ) |∅i . (2.29)In the second step, the extra vertex operators Γ ± ( q − ρ ) are introduced on the left/right of thecorrelators. Since these operators are built purely upon either positive or negative modes, theygive no extra contribution. Based on our previous discussion, we now understand that the reasonbehind this clever trick is to reconstruct the vacuum component of the intertwiner, Z ( q, s, t ) = h∅| Φ ∅ ρ ( D ) q − s ,v ( e H ( t ) ) Q L Φ ∅ |∅i . (2.30)It allows us to use the exchange relation 2.20 for the exponential of the evolution operator H ( t ). ρ ( D ) − u, ( T · e H ( t ) )Φ ∅ = Φ ∅ ρ ( D ) u, ( e H ( t ) ) . (2.31)Computing the action of the automorphism T on these operators with the help of the equation 2.2,we can rewrite the amplitude as Z ( q, s, t ) = e P k> t k qk − qk h∅| ρ ( D ) − q − s , (cid:16) e P k> t k q − k / W k, − k (cid:17) Φ ∅ Q L Φ ∅ |∅i . (2.32)The last step consists in transforming the generators W k, − k into the Heisenberg modes J k = ¯ W k, .It is done using the framing operators since the specialization of the formula 2.15 at α = 1 and( m, n ) = ( k, − k ) reads q ˜ W ρ ( D ) u, ( W k, − k ) q − ˜ W = u − k q k / J k . The framing operator can be introducedon the left/right of the correlator for free since its action on the vacuum state is trivial. In this way,we find the desired result Z ( q, s, t ) = e P k> t k qk − qk h∅| e P k> ( − q s ) k t k J k g |∅i , g = q ˜ W Φ ∅ Q L Φ ∅ q ˜ W . (2.33)Then, the fact that Z ( q, s, t ) is a tau function of the KP hierarchy follows from the fact that g ⊗ g commutes with Ψ = P r ψ r ⊗ ¯ ψ − r . This type of operators were called the group-like elements of10 L ( ∞ ) in [7]. This property is sufficient to ensure that Z ( q, s, t ) obeys the Hirota equation. Let usstress that the bosonic formula for Φ ∅ and the fermionic realization of the framing operator through2.12 are essential to show the group-like property. The fact that the hierarchy is of KP-type followsfrom the possibility to move the time dependency to the right of the correlators. We refer to theoriginal papers [1, 47, 48] for a more thorough discussion that also includes the Lax formalism.We conclude this section with a short remark. One may wonder what would happen if we considerthe other vertical components Φ ν of the intertwining operator instead of the vacuum component.In this case, one can show that the exchange relation 2.20 is satisfied only for the generators W m,n with the index n < | ν | . As a consequence, we need to restrict ourselves to the times t k with k > | ν | .Replacing Φ ∅ with Φ ν in the operator g and turning off the lower times, we find another tau functionthat reads Z ν ( q, s, t ) = t ν X λ s λ ( q − ρ − ν ) s λ ( q − ρ − ν ′ ) Q | λ | e P k> | ν | t k φ k ( λ,s ) . (2.34)This tau function appears to be a particular case of the generating functions considered in [51]. Itis naturally associated to the topological vertex C ∅ , ∅ ,ν that enumerates plane partitions with fixedasymptotics ν in one direction [40]. In this section, we attempt to generalize the construction of a tau function to the refined meltingcrystal introduced in the context of topological strings in [15]. The main idea is to consider acounting function of plane partitions where boxes have a different weight q = q, t depending ontheir location, Z ( t, q ) = X π Y ∈ π q = ∞ Y i,j =1 (1 − t i q j − ) − . (3.1)Using the Cauchy identity, the double product can be written as a sum over Macdonald polynomials, Z ( t, q ) = X λ ( P λ ( t − ρ )) h P λ , P λ i q,t , (3.2)it reduces to the expression 2.27 of Z ( q ) in the limit t = q since Macdonald polynomials reduceto Schur polynomials and their norm 2.7 tend to one. However, in order to be able to employ theintertwiner Φ( v ), instead of its dual Φ ∗ ( v ), it is more convenient to generalize the formulas 2.27 as Z ( t, q ) = X λ ( ιP λ ( t − ρ )) h P λ , P λ i q,t → Z ( Q, t , t, q ) = X λ Q | λ | ( ιP λ ( t − ρ )) h P λ , P λ i q,t e P k> t k Φ k ( λ,u ) , (3.3) We refer to the appendix A of ref. [15] for the exact prescription. ι : p k ( x ) → − p k ( x ) acting on the ring of symmetric polynomials byreversing the sign of elementary power sums. In order to define properly the deformed quantitiesΦ k ( λ, u ), we need to generalize the algebraic description to the refined case. The relevant algebra isthe quantum toroidal algebra of gl (1), it depends on two parameters ( q , q ) that are identified withthe parameters ( q, t − ) of the Macdonald polynomials. It reduces to the quantum W ∞ algebrain the limit t → q which corresponds to the self-dual limit of the omega-background ε + ε → F , called the Fock (or horizontal)representation [52, 53], which we will review shortly below.To obtain the bosonic expression 1.1 for the amplitude Z ( Q, t , t, q ), two main ingredients areneeded: an exchange relation generalizing 2.31 and a framing operator that will play the role of q ˜ W . Like before, the exchange relation will be obtained as a projection of the AFS intertwiningrelation [11] on the vacuum component of the intertwiner in the vertical direction. On the otherhand, the framing operator will be constructed by considering the refined framing factors [15, 54].We will then show that this operator obeys an equivalent of the algebraic property 2.15. gl (1) algebra We review briefly here the definition of the quantum toroidal gl (1) algebra. We mostly follow thenotations and conventions of [32], and refer to [27, 42, 43, 55] for more details on the correspondencewith the ( p, q )-brane construction of topological strings amplitudes.The algebra is usually formulated in terms of four Drinfeld currents, x ± ( z ) = X k ∈ Z z − k x ± k , ψ ± ( z ) = X k ≥ z ∓ k ψ ±± k . (3.4)They satisfy a set of exchange relations that can be found, e.g. in [27,32], but we prefer to work heredirectly with the modes x ± k , ψ ±± k . The subalgebra generated by the elements ψ ±± k is the analogueof the Cartan subalgebra of quantum affine algebras, it has an alternative formulation in terms ofmodes a k defined by exponentiation, ψ ± ( z ) = ψ ± exp ± X k> z ∓ k a ± k ! , (3.5)and satisfying a twisted Heisenberg algebra. The algebra has only two parameters ( q , q ), but itis useful to introduce a third one through the relation q q q = 1. We also introduce the shortcut Both P λ ( x ) and ιP λ ( x ) have the same Cauchy identity, which provides two different ways of writing the doubleinfinite product 3.1. Note also that in the limit t = q , ιP λ ( x ) → ( − | λ | s λ ′ ( x ) and the first equality in 3.3 reproducesagain the expression 2.27 of Z ( q ) upon the replacement λ → λ ′ in the summation. γ = q / = ( q q ) − / . The algebra has two central charges ( c, ¯ c ), the second one enteringthrough the zero modes of the Cartan currents ψ ± = γ ∓ ¯ c . The modes of the currents satisfy thecommutation relations[ a k , a l ] = ( γ kc − γ − kc ) c k δ k + l , [ a k , x ± l ] = ± γ ∓| k | c/ c k x ± l + k , [ x + k , x − l ] = κγ ( k − l ) c/ ψ + k + l , k + l > κγ ( k − l ) c/ ψ +0 − κγ − ( k − l ) c/ ψ − , k + l = 0 − κγ − ( k − l ) c/ ψ − k + l , k + l < , (3.6)with the coefficients κ = (1 − q )(1 − q )(1 − q q ) , c k = − k Y α =1 , , (1 − q kα ) . (3.7)They form a Hopf algebra with the Drinfeld coproduct∆( x + k ) = x + k ⊗ X l ≥ γ − ( c ⊗ k + l/ ψ −− l ⊗ x + k + l , ∆( x − k ) = X l ≥ γ − (1 ⊗ c )( k − l/ x − k − l ⊗ ψ + l + 1 ⊗ x − k , ∆( a k ) = a k ⊗ γ −| k | c/ + γ | k | c/ ⊗ a k , ∆( c ) = c ⊗ ⊗ c, ∆(¯ c ) = ¯ c ⊗ ⊗ ¯ c. (3.8) Automorphisms
The algebra is known to possess the group of automorphisms SL(2 , Z ) generatedby the elements S and T . The action of the automorphism T on the modes of the Drinfeld currentscan be expressed easily: it leaves the Cartan modes a k invariant and acts as T · x ± k = x ± k ∓ , T · ( c, ¯ c ) → ( c, ¯ c + c ) . (3.9)The automorphism S has been uncovered by Miki in [13]. It is of order four, and is defined uniquelyby its action on the modes x ± , a ± , namely a → ( γ − γ − ) x +0 → − a − → − ( γ − γ − ) x − → a , (3.10)and the central elements ( c, ¯ c ) → ( − ¯ c, c ). The explicit transformation formulas of the modes x ± k and ψ ±± k are useful here, but, since they are more complicated, we decided to confine them to theappendix A to avoid introducing too many notations. Let us only define the notation b k = S · a k for the S -dual Cartan modes. Self-dual limit
In the self-dual limit ( q , q ) → ( q, q − ), the modes of the quantum toroidal gl (1)algebra satisfy the commutation relations of the quantum W ∞ algebra. The identification goesas follows, x + k − q → q k/ W k, + δ k, c − q , x − k − q → q − k/ W k, − + δ k, c − q − ,ka k ( q k/ − q − k/ )( q k/ − q − k/ ) → W k, − c − q k , ( c, ¯ c ) → ( c , − c ) . (3.11)13hus, the roles of W k, , W ,k and W k, − k is played in the refined case by a k , b − k and T · b − k respectively. The role previously devoted to the Dirac representation ρ ( D ) u,v will now be played by the horizontalrepresentation [52]. This representation has also the levels (1 ,
0) and acts on the free boson Fockspace F . It has a weight u ∈ C × and will be denoted ρ (1 , u . In this representation, the Drinfeldcurrents take the form of vertex operators defined upon the Heisenberg modes J k representing theCartan modes a k , ρ (1 , u ( a k ) = γ k/ k ( γ k − γ − k )(1 − q k ) J k , ρ (1 , u ( a − k ) = γ k/ k ( γ k − γ − k )(1 − q k ) J − k , ( k > ,ρ (1 , u ( x ± ( z )) = u ± exp ± X k> z k k γ (1 ∓ k/ (1 − q k ) J − k ! exp ∓ X k> z − k k γ (1 ∓ k/ (1 − q k ) J k ! . (3.13)Using the isomorphism between F and the ring of symmetric polynomials, the action of the modes a k on the Macdonald states | P λ i corresponds to either a multiplication by the power sums p k or thederivation ∂/∂p k depending on the sign of k . Since the power sum p coincides with the elementarysymmetric polynomial e , the action of a ± is deduced from the Pieiri rules obeyed by Macdonaldpolynomials, ρ (1 , u ( a ) | P λ i = X ∈ R ( λ ) r ( − ) λ ( ) | P λ − i , ρ (1 , u ( a − ) | P λ i = X ∈ A ( λ ) r (+) λ ( ) | P λ + i . (3.14)We denoted here A ( λ ) and R ( λ ) the sets boxes that can be added to/removed from the Youngdiagram of λ . The coefficients r ( ± ) λ ( ) depend on the choice of normalization for the modes andstates, they can be computed explicitly but we will not need their expression here.Finally, using Miki’s automorphism, we can also determine the action of the modes b k on theMacdonald states [32]. It is used here to define the quantities Φ k ( λ, u ) coupled to the times t k , ρ (1 , u ( b − k ) | P λ i = Φ k ( λ, u ) | P λ i , Φ k ( λ, u ) = ( − k γ k u − k c k X ( i,j ) ∈ λ q − ( i − k q − ( j − k − − q − k )(1 − q − k ) | P λ i . (3.15) These modes correspond to the three patches in [34], as can be seen from their fermionic realization¯ W k, = I dz iπ z k : ¯ ψ ( z ) ψ ( z ) : , ¯ W ,k = I dz iπ : ¯ ψ ( z ) ψ ( e kg str z ) : , ¯ W k, − k = I dz iπ z k : ¯ ψ ( z ) ψ ( e − kg str z ) : . (3.12) P λ . In the self-dual limit, k Φ k ( λ, q − s )( q k/ − q − k/ )( q k/ − q − k/ ) → − ( − k q − k/ (cid:20) φ k ( λ, s ) + 11 − q − k (cid:21) , (3.16)and the deformed amplitude Z ( Q, t , t, q ) reduces to the tau function 2.33 (up to a rescaling of thetimes parameters). Note, however, that the second term in the bracket is responsible for an extraexponential factor in the formula 2.33, it will not be present in the refined case. The intertwiner constructed by Awata, Feigin and Shiraishi in [11] intertwines the representation oflevels (1 , n + 1) and the tensor product of two representations with levels (0 ,
1) and (1 , n ). Here, weonly need to consider the case n = 0. Moreover, the representations of levels (1 ,
1) and (0 ,
1) canbe obtained from the horizontal representation (1 ,
0) using the automorphisms S and T describedabove. From the analysis of the transformation of representations performed in [32], the AFSintertwining equation can rewritten in the form ρ (1 , uv ( T · e )Φ( v ) = Φ( v ) (cid:0) ρ (1 , v ◦ S ⊗ ρ (1 , u ∆( e ) (cid:1) , (3.17)for any element e of the quantum toroidal gl (1) algebra. The solution of this equation has beenfound in [11], it can be expanded over the vertical components as a sum of vertex operators,Φ( v ) = X λ Φ λ ( v ) hh P λ | , Φ λ ( v ) = t λ : Φ ∅ ( v ) Y ( i,j ) ∈ λ η ( vq i − q j − ) : , with Φ ∅ ( v ) = exp − X k> v k k (1 − q k ) J − k ! exp X k> q − k v − k k (1 − q k ) J k ! . (3.18)The operator η ( z ) = ρ (1 , ( x + ( z )) coincides with the representation of the current x + ( z ) given in3.13. The normalization factor t λ is not important here as we only consider the vacuum component,and we can always set t ∅ = 1. In the melting crystal formalism, the vacuum component correspondsto Φ ∅ ( v ) = Γ − ( vt / − ρ )Γ + ( v − γ − q − / − ρ ) . (3.19)To derive the exchange relation, we exploit the fact that S · x − k annihilates the vacuum state |∅ii in the vertical channel. This fact follows from the application of Miki’s automorphism to mapthe vertical representation of levels (0 ,
1) to the horizontal one, since the vertical action of x − k annihilates the vacuum [32]. However, to derive an exchange relation for the modes b − k , there is15mall difficulty coming from the fact that their coproduct involves an infinite sum. For instance,for b − = ( γ − γ − ) x − , we have( γ − γ − ) − ∆( b − ) = X k ≥ γ k (1 ⊗ c ) / x −− k ⊗ ψ + k + 1 ⊗ x − . (3.20)Here, the AFS equation 3.17 does simplify into an exchange relation because the vertical actionof all the terms x −− k ⊗ ψ + k vanishes. After the projection of the resulting equation on the verticalvacuum component, we find ρ (1 , uv ( T · b − k )Φ ∅ ( v ) = Φ ∅ ( v ) ρ (1 , u ( b − k ) (3.21)for k = 1. The proof for higher k > H ( t ) = P k> t k b − k , ρ (1 , uv ( T · e H ( t ) )Φ ∅ ( v ) = Φ ∅ ( v ) ρ (1 , u ( e H ( t ) ) ,ρ (1 , uv ( e H ( t ) )Φ ∅ ( v ) = Φ ∅ ( v ) ρ (1 , u ( T − · e H ( t ) ) . (3.22)The second exchange relation is also derived in appendix A, it is obtained from the the AFS relation3.17 applied to e = T − · b − k To avoid cluttering the formula, we omit the dependence in the variables ( q, t ) in this section. Inorder to construct the bosonic expression for the amplitude Z ( Q, t ) defined in 3.3, we need theexpression of the matrix elements of the intertwiner Φ( v ) in the Macdonald basis, h∅| Φ ∅ ( q / v ) | P λ i = v −| λ | γ −| λ | ιP λ ( t − ρ ) , h P λ | Φ ∅ ( q / v ) |∅i = v | λ | γ −| λ | ιP λ ( t − ρ ) . (3.23)These formulas are used to replace the Macdonald polynomials in 3.3 with bosonic correlators.Then, the time dependence is produced using the diagonal action 3.15 of the modes b − k on theMacdonald basis. Finally, the summation of the Young diagrams λ is performed using the closurerelation of the Macdonald basis, and we find Z ( γ − Q, t ) = h∅| Φ ∅ ( v ) ρ (1 , u ( e H ( t ) )Φ ∅ ( Qv ) |∅i . (3.24)Note that this quantity is actually independent of v , and the dependence in u can be eliminatedby a rescaling of the times t k → u k t k . The second intertwiner can be replaced by Φ ∅ ( Qv ) = Since p k ( t ± ρ ) = ± / ( t k/ − t − k/ ) with | t | ± > P λ ( t ρ ) → ιP λ ( t − ρ ) in theexpression (4.13) of ref. [11] when | t | < L Φ ∅ ( v ) Q − L where L is the Fock space operator introduced in 2.12. Since Macdonald polyno-mials, just like Schur polynomials, are homogeneous polynomials of degree | λ | , it acts diagonally onthe Macdonald basis, L | P λ i = | λ | | P λ i . We proceed to move the time dependence to the left. The first step is performed using theexchange relation 3.22, it gives Z ( γ − Q, t ) = h∅| ρ (1 , uv ( T · e H ( t ) )Φ ∅ ( v )Φ ∅ ( Qv ) |∅i . (3.26)For the next step, we need to define the framing operator. In the refined case, the framing factor ismodified into f λ = q n ( λ ′ ) t − n ( λ ) with n ( λ ) = P ( i,j ) ∈ λ ( i −
1) and n ( λ ′ ) = P ( i,j ) ∈ λ ( j − F as a diagonal operator on the Macdonald states, with eigenvalues F | P λ i = F λ | P λ i , F λ = Y ( i,j ) ∈ λ q i − q j − = q n ( λ )1 q n ( λ ′ )2 . (3.27)In the limit t = q , this operator tends to q − W / , the shift L is missing because we treat the v -dependence differently here. Like the operator q ˜ W in the self-dual case, the operator F is deeplyconnected to the automorphism T , i.e. ρ (1 , u ( T · b − k ) = ( − k +1 u − k γ k/ F ρ (1 , u ( a k ) F − , k ∈ Z ,ρ (1 , u ( T − · b − k ) = ( − k u − k γ k/ F − ρ (1 , u ( a − k ) F, k ∈ Z . (3.28)The derivation of these two identities is a bit involved as it makes use of Miki’s automorphism, itis presented in appendix A. Using the first identity, and the horizontal representation 3.13 of themodes a k , we can write the partition function in the form Z ( γ − Q, t ) = h∅| e P k k τ k J k g |∅i , with g = F − Φ ∅ ( v )Φ ∅ ( Qv ) F − , (3.29)and the rescaled times τ k = ( − uv ) − k (1 − q k )(1 − q k ) t k . This equation is the main result of thissection. Unfortunately, we were not able to show that the operator g is a group like element of GL ( ∞ ). Group like elements form a monoid, and we can examine the decomposition of g into itselementary factors. The vacuum component of the intertwiner is still a group like element, despitethe asymmetry between positive and negative modes as any operator of the form : e P k ∈ Z × t k J k : isgroup-like. On the other hand, we could not show that F is group-like, and strongly suspect that itis not from a perturbative analysis of the Hirota equation. As we shall explain in the next section,it is likely that the operator Ψ has to be replaced by a different operator. We can also see L as the representation of the grading operator d for the quantum toroidal algebra (see [32]) ρ (1 , u ( d ) = X k> J − k J k = X r ∈ Z +1 / r : ¯ ψ − r ψ r := L . (3.25)
17e conclude with another remark. Using the second exchange relation in 3.22 together with thesecond identity in 3.28, the exponential of H ( t ) can be moved to the right in the correlator instead, Z ( γ − Q, t ) = h∅| ge − P k> ¯ τ k J − k |∅i , (3.30)with ¯ τ k = ( − u/Qv ) − k (1 − q k )(1 − q k ) t k . Unlike in the case of the KP hierarchy, the times variablesare not equal but they obey a simple scaling relation,¯ τ k = 1 − q k − q k Q k v k τ k . (3.31) Our main result is the observation of a relation between the tau function of an integrable hierarchy,the intertwining operator of a quantum algebra, and the framing operator. The fundamental roleof the SL(2 , Z ) group of automorphisms in this description has been emphasized as it is deeplyrelated to both intertwiner and framing operator. Our observation offers the possibility to extendthe correspondence between topological strings theory and integrable hierarchies in several newdirections. The most obvious one is to consider more involved toric diagrams by exploiting thegluing rules of the topological vertex [27, 42, 43]. The next simplest toric diagram describes theresolved conifold and the corresponding time-deformed amplitude was shown to be a tau functionof a different reduction of the Toda hierarchy called the Ablowitz-Ladik hierarchy [48, 56]. Thisresult could be reproduced within our algebraic formalism provided that we introduce the dualintertwiner Φ ∗ , also constructed in [11], that is expected to enjoy a similar exchange relation.The trinion theories T N provide another set of interesting toric diagrams [57–59]. The algebraicobject obtained by gluing intertwiners according to these diagrams is also an intertwiner but itinvolves representations of the quantum toroidal algebra with higher levels, namely (
N, N ), ( N, , N ). The intertwining relation projected on the vertical component is expected to producean exchange relation similar to 3.22. Furthermore, the framing operator can be easily generalizedto representations (0 , N ) by taking the tensor product of the (0 ,
1) framing operator defined inthis paper. Thus, our construction should apply in this case as well, and the corresponding time-deformed amplitude should be the tau function of an integrable hierarchy in the self-dual limit.The second part of the paper is an attempt to define a refined tau function from the naturaldeformation of the algebraic objects, lifting them from the quantum W ∞ algebra to the quantumtoroidal gl (1) algebra. We ended up with the bosonic expression 3.29 for the refined amplitude, butwe were unable to show that the refined framing operator entering in the operator g is a group-like element for GL ( ∞ ). Since the Dirac fermion plays no role in the representation theory of the The author would like to thank E. Pomoni for drawing his attention to this family of theories. P r ψ r ⊗ ¯ ψ − r is replaced by adifferent operator, just like in the Kac-Wakimoto construction. At the moment, it is not clear whatthis operator should be. In this respect, the Kac-Wakimoto construction for the toroidal algebrarealized in [60] might be a good source of inspiration.As an intermediate step in the deformation of the fermionic structure, one could focus on the q = 0 limit in which Macdonald polynomials reduce to Hall-Littlewood polynomials. In [61], Jingintroduced certain vertex operators as a t -deformation of the Dirac fermion. These operators mightbe used to deform the Hirota equation, or the Lax formalism.The algebraic description of topological string theory has been extended to different algebrasand geometric backgrounds. Some of these algebras should possess an SL(2 , Z ) subgroup of auto-morphisms, like the quantum toroidal gl ( p ) algebras [18], their elliptic deformations [19, 62] or eventhe fully deformed algebra of [23]. In all these cases, we expect our construction to apply, producingtau functions of different integrable hierarchies in specific limits.Going in the other direction, one might try to build a vertex operator from known integrablehierarchies. For instance, the quantum algebra associated to BKP, CKP and DKP hierarchies [6,63]are known to be orbifolds of the quantum W ∞ algebra [39]. This approach should meet with theearlier attempt of Foda and Wheeler to build a B-type topological vertex [64, 65]. We hope to beable to report on this problem soon.Finally, we have been working here with the A-model formulation of topological strings. Inthe B-model, the connection with the KP hierarchy can also be seen using the Hermitian matrixmodel. In this context, the refinement is well-understood as the matrix model is replaced by abeta-ensemble [66]. It would be instructive to reproduce our derivation on the other side of themirror.
Acknowledgements
The author would like to thank Sasha Alexandrov, Yutaka Matsuo, Elli Pomoni and KanehisaTakasaki for very helpful discussions. He is also very grateful to Pr. Kimyeong Lee and the KoreaInstitute for Advanced Study (KIAS) for their generous support in these difficult times. Note that the fermionic description is lost for both A and B models after the refinement. Proofs
A.1 Reminder on Miki’s automorphism
This reminder is a brief summary of the appendix A in [32] that is itself based on the originalpaper [13]. We denote by y ± ( z ) = S · x ± ( z ) and ξ ± ( z ) = S · ψ ± ( z ) the image of the Drinfeldcurrents under Miki’s automorphism. Just like the original currents in 3.6, the S-dual currents canbe decomposed in terms of modes y ± ( z ) = X k ∈ Z z − k y ± k , ξ ± ( z ) = X k ≥ z ∓ k ξ ±± k = ξ ± exp ± X k> z ∓ k b ± k ! , (A.1)with y ± k = S · x ± k , b k = S · a k and ξ ±± k = S · ψ ±± k . Since S is an automorphism, these new modessatisfy the same commutation relations as the original algebra, e.g.[ b k , b l ] = − ( γ k ¯ c − γ − k ¯ c ) c k δ k + l , [ b k , y ± l ] = ± γ ±| k | ¯ c/ c k y ± l + k , [ y + k , y − l ] = κγ − ( k − l )¯ c/ ξ + k + l , k + l > κγ − ( k − l )¯ c/ ξ +0 − κγ ( k − l )¯ c/ ξ − , k + l = 0 − κγ ( k − l )¯ c/ ξ − k + l , k + l < . (A.2)The expression for the S-dual modes y ± k , ξ ±± k in terms of the original ones has been obtained byMiki in [13], y ± k = ( ± ) k γ − ( c ± k ¯ c ) / σ − ( k − (cid:16) ad x +0 (cid:17) k − x + ∓ , y ±− k = − ( ± ) k γ ( c ∓ k ¯ c ) / σ − ( k − (cid:16) ad x − (cid:17) k − x −∓ ,ξ ±± k = − ( ∓ ) k ( γ − γ − ) σ − ( k − γ ∓ c ad x ±∓ (cid:16) ad x ± (cid:17) k − x ±± , ξ ±± = ± γ ∓ c ( γ − γ − ) x ± , ξ ± = γ ∓ c . (A.3)In these formulas, we denoted the adjoint action ad A B = [ A, B ] and σ = ( q / − q − / )( q / − q − / ). A.2 Proof of the refined exchange relations
In order to prove the first exchange relation 3.22, we use the following fact: since ρ (1 , is a repre-sentation and T an automorphism, if two elements satisfy the exchange relation, so does their sum,product, commutator,... We have already shown that b − ∝ x − obeys the exchange relation. Inthe same way, it is possible to show that the AFS relation 2.16 with e = x − k produces an exchangerelation of the type 3.22 where b − k is replaced by x − k . It follows from the coproduct 3.8 and thefact that the modes x − k annihilate the vacuum in the vertical representation of levels (0 , c obeys the exchange relation but ¯ c does not since T · ¯ c = c + ¯ c . Since the expressionA.3 for ξ −− k involves only sums and products of x − k and c , this operator obeys the exchange relation20nd so does b − k for k > z . Note, however, that b ∝ x +0 does not satisfy theexchange relation, and neither does the modes b k for k >
0. The second exchange relation holdsbecause the action of T − simply shifts the modes x − k → x − k +1 in the expression of ξ −− k and thus theprevious arguments hold as well in this case. A.3 Proof of the algebraic properties for the refined framing operator
We present here the proof of the first identity in 3.28 for k >
0. The other cases, namely k < F acted on by the representation ρ (1 , u , we omit to indicate the representation of the quantum toroidal modes. We start with the case k = 1. Once combined with Miki’s transformation A.3, the mode expansion A.1 for ξ − ( z ) gives atfirst orders in z , b − = ( γ − γ − ) x − ⇒ T · b − = ( γ − γ − ) x − . (A.4)Using the algebraic relations 3.6, this can be written further T · b − = − γ − c/ c − ( γ − γ − )[ a , x − ] . (A.5)The action of the r.h.s. on the Macdonald states can be computed explicitly using the Pieri rules3.14 for a and the fact that x − ∝ b − is diagonal (see 3.15), a | P λ i = X ∈ R ( λ ) r ( − ) λ ( ) | P λ − i , x − | P λ i = Φ ( λ, u ) γ − γ − | P λ i . (A.6)We do not need the explicit expression for the coefficients r ( − ) λ ( ). Then, denoting χ = q i − q j − the content of a box = ( i, j ) ∈ λ , we compute T · b − | P λ i = − γ − / c − X ∈ R ( λ ) r ( − ) λ ( ) (Φ ( λ, u ) − Φ ( λ − , u )) | P λ − i = u − γ / X ∈ R ( λ ) r ( − ) λ ( ) χ − | P λ − i = u − γ / F a F − | P λ i . (A.7)Thus, we have shown the identity 3.28 for k = 1. To simplify the upcoming formulas, we introducethe rescaled mode α = a / ( γ − γ − ), so the previous identity writes x − = u − γ / F α F − .Before addressing the general case, we would like to start with a short remark. Using thealgebraic relations 3.6, it is possible to write down ψ + k = κ − γ kc/ [ x +0 , x − k ] , x − k = ( − k γ − kc/ c − k (ad a ) k x − . (A.8)21ombining both, we arrive at an expression that be useful later, ψ + k = κ − ( − k c − k ad x +0 (ad a ) k x − . (A.9)In a similar way, we write the modes ξ −− k = − κ − γ ( k +2)¯ c/ [ y + − ( k +1) , y − ] for k > y ± k using the relations A.2, and then use Miki’s transformation A.3 to write down y + − ( k +1) = − γ c/ γ − ( k +1)¯ c/ σ − k (ad x − ) k x −− , y − = − γ − ( c − ¯ c ) / x +1 . (A.10)As a result, we find ξ −− k = κ − γ ¯ c σ − k ad x +1 (ad x − ) k x −− ⇒ T · ξ −− k = κ − γ c +¯ c σ − k ad x +0 (ad x − ) k x − . (A.11)Using the identity found previously for x − , and the fact that x − commutes with F (they are bothdiagonal in the Macdonald basis), we have e zu ad x − x − = e zux − x − e − zux − = F e zγ / α x − e − zγ / α F − = F e ad zγ / α x − F − . (A.12)Expanding in powers of z , we deduce that T · ξ −− k = κ − γ k/ σ − k u − k ad x +0 (cid:0) F (cid:0) (ad α ) k x − (cid:1) F − (cid:1) , (A.13)where we have also identified the central charges ( c, ¯ c ) with the levels (1 , A, F BF − ] = F [ F − AF, B ] F − , together with the fact that x +0 also commutes with F , to write down T · ξ −− k = κ − γ k/ σ − k u − k F (cid:16) ad x +0 (ad α ) k x − (cid:17) F − . (A.14)Comparing with the formula A.9 for the modes ψ + k , we find that T · ξ −− k = γ k/ ( − u ) − k F ψ + k F − ⇒ T · ξ − ( z ) = γF ψ + ( − γ − / uz − ) F − . (A.15)The first identity in 3.28 with k > γ . Applying the same method to T · ξ + ( z ),we can prove the following identities that produce the other cases of the relations 3.28 by expansion, T · ξ + ( z ) = γ − F ψ − ( − γ − / uz − ) F − , T − · ξ − ( z ) = γF − ψ − ( − γ / u − z ) F, T − · ξ + ( z ) = γ − F − ψ + ( − γ / u − z ) F. (A.16)22 eferences [1] T. Nakatsu and K. Takasaki, “Melting crystal, quantum torus and Toda hierarchy,” Commun. Math. Phys. (2009) 445–468, arXiv:0710.5339 [hep-th] .[2] M. Sato,“Soliton Equations as Dynamical Systems on Infinite Dimensional Grassmann Manifold,” in
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