Quantum W_{1+\infty} subalgebras of BCD type and symmetric polynomials
aa r X i v : . [ h e p - t h ] J a n Quantum W ∞ subalgebras of BCD typeand symmetric polynomials Jean-Emile Bourgine
Korea Institute for Advanced Studies (KIAS)Quantum Universe Center (QUC)85 Hoegiro, Dongdaemun-gu, Seoul, South Korea [email protected]
Abstract
The infinite affine Lie algebras of type ABCD, also called b gl ( ∞ ), b o ( ∞ ), b sp ( ∞ ), are equiv-alent to subalgebras of the quantum W ∞ algebras. They have well-known representationson the Fock space of either a Dirac fermion ( ˆ A ∞ ), a Majorana fermion ( ˆ B ∞ and ˆ D ∞ ) or asymplectic boson ( ˆ C ∞ ). Explicit formulas for the action of the quantum W ∞ subalgebrason the Fock states are proposed for each representation. These formulas are the equivalentof the vertical presentation of the quantum toroidal gl (1) algebra Fock representation. Theyprovide an alternative to the fermionic and bosonic expressions of the horizontal presentation .Furthermore, these algebras are known to have a deep connection with symmetric polynomi-als. The action of the quantum W ∞ generators leads to the derivation of Pieri-like rulesand q-difference equations for these polynomials. In the specific case of ˆ B ∞ , a q-differenceequation is obtained for Q -Schur polynomials indexed by strict partitions. edicated to the memory of Omar Foda. The Lie algebra b gl ( ∞ ) is usually defined as a central extension of the algebra gl ( ∞ ) of matrices withinfinite size [1]. Through its well-known representation on the Dirac fermion Fock space, it has beeninstrumental in finding the solutions of the Kadomtsev-Petviashvili (KP) hierarchy [2]. Moreover,it also shares a deep relation with Schur symmetric polynomials that provide tau functions forthe hierarchy. Its subalgebras of BCD type, namely ˆ B ∞ , ˆ C ∞ and ˆ D ∞ , have similar definitions,with further symmetry constraints imposed on the infinite matrices. They are naturally associatedto the integrable hierarchies of type BKP, CKP and DKP [3, 4]. The BKP and DKP hierarchieshave polynomial tau functions given by the Q-Schur polynomials [5, 6], a specialization of the Hall-Littlewood polynomials at t = − On the other hand, it was shown in [9] that the CKPhierarchy does not admit any polynomial tau function. Yet, symmetric polynomials associated tothe ˆ C ∞ algebra have been introduced recently by van de Leur, Orlov and Shiota in [10], and wewill review their construction below. Most of these works focus on the important application tointegrable hierarchies, but we would like to revisit here the relation between infinite Lie algebrasand symmetric polynomials from a different perspective, namely quantum algebras.As the name suggests, the quantum W ∞ algebra is a q-deformation of the W ∞ algebra[11, 12]. The algebra depends on a parameter q ∈ C × , and is defined in terms of the generators W m,n , with indices ( m, n ) ∈ Z × Z \ { (0 , } , and a central element C , satisfying the commutationrelations [ 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 ′ (cid:19) . (1.1)Here, we assume that q is not a root of unity, and denote the algebra W for short. This algebracan be realized in terms of b gl ( ∞ ) generators using a linear relation reminiscent of a discrete Fouriertransform (see the equation 2.2 below). Subalgebras W X for X = B, C, D can be defined as a quo-tient of W by the group of automorphisms Z = { , σ X } generated by the involutive automorphism σ X defined below. In the context of W-algebras, this procedure is called orbifolding and we will usethis terminology here. These subalgebras are realized in terms of the generators of the subalgebrasˆ X ∞ of b gl ( ∞ ) in the same way W is [13, 14].The quantum toroidal algebra of gl (1), or Ding-Iohara-Miki algebra [15,16], is a refinement of thequantum W ∞ algebra. Following the inspirational paper [18], the former has played an important Instead, the specialization at t = 0 and t = 1 gives respectively the Schur polynomials and the symmetricmonomials m λ . When n + n ′ = 0, the central term becomes q − mn mCδ m + m ′ . The quantum toroidal algebra of gl (1) depends on two parameters q , q , in the limit q q → W plays thesame role as the quantum toroidal gl (1) algebra in the self-dual limit of topological strings [17], andthat it can be used in the same way to build the original topological vertex [38], thus providing thequantum algebraic framework behind the melting crystal construction of Okounkov, Reshetikhinand Vafa [39].Given the observed connections between topological strings and integrable hierarchies [38,40–42],the relation between W and b gl ( ∞ ) comes as no surprise. One of the motivation for this paper isto extend these connections to the subalgebras W X . For this purpose, it appeared necessary torevisit first the relations between algebras and symmetric polynomials with a focus on the quantum W ∞ generators. To be specific, we examine the Fock representations of the algebras ˆ A ∞ , ˆ B ∞ , ˆ C ∞ ,ˆ D ∞ , and derive explicit expressions for the action of the generators on the Fock states. We furtherassociate to each algebra a set of symmetric polynomials indexed by the states defining the basis ofthe representation. Then, the action of the W X -algebra induces identities among the polynomials:Pieri-like relations or q-difference equations.Hence, we examine the level one representation of the b gl ( ∞ ) algebra on the Fock space of a2D Dirac fermion. The states of the PBW basis are labeled by partitions, they are in one-to-onecorrespondence with the ring of symmetric polynomials in infinitely many variables. We deriveexplicit combinatorial formulas for the action of the generators W m,n on the states associated toSchur polynomials. From these expressions, we recover the rules obeyed by the Schur polynomialsunder multiplication by elementary power sums, together with the q-difference equation comingfrom the specialization of the Macdonald operator at t = q [8]. Expanding this operator in q , werecover the Hamiltonians of the Calogero-Sutherland integrable model [43]. This program is thenextended to the subalgebras ˆ B ∞ , ˆ C ∞ and ˆ D ∞ with a mitigated success. In the case of ˆ B ∞ , westudy the representation of level 1 / Q -Schur functions, and the whole program goes through. Inparticular, we obtain a new q-difference equation solved by these polynomials. On the other hand,the definition of symmetric polynomials is not obvious in the case of the algebras ˆ C ∞ and ˆ D ∞ . Thecorresponding representations have levels C = − / C = 1 / A ∞ and ˆ B ∞ ,and we study the properties of both sets of symmetric polynomials. version of the algebra W with an extra central element C ′ [17]. λ = (4 , , , ,
1) with three hooks of length 8 , , A ∞ , ˆ B ∞ , ˆ C ∞ and ˆ D ∞ . The first appendix re-derive the bosonicexpression of the Schur polynomials, while the second appendix focuses on the derivation of theq-difference equations. Notations
In this paper, we take the convention that indices i, j, k, l, m, n take integer values,while indices r, s, t take half-integer values. Moreover, partitions are denoted with the Greek letters λ, µ, ν , they consists in sets of ordered positive integers λ = ( λ ≥ λ ≥ · · · ) with | λ | = P i λ i finite.They are represented as Young diagrams with columns of λ i boxes, drawn upward (see the figure1). We denote ℓ ( λ ) = ♯ { λ i = 0 } the number of parts, | λ | the number of boxes and d ( λ ) the numberof hooks. W ∞ and b gl ( ∞ ) The algebra b gl ( ∞ ), or ˆ A ∞ , can be formulated in terms of the generators E r,s with half-integerindices, and a central element C , obeying the commutation relations,[ E r,s , E t,u ] = δ s + t E r,u − δ r + u E t,s + C ( θ ( r ) − θ ( t )) δ s + t δ r + u , r, s, t, u ∈ Z + 12 , (2.1)where θ ( r ) denotes the Heaviside function taking the value one for r > r < ρ ( E r,s ) = e r +1 / , − s +1 / where e i,j ( i, j ∈ Z ) are the matrices with1 at position ( i, j ) and zero elsewhere that form a basis of gl ( ∞ ).The algebraic relations 1.1 defining the quantum W ∞ algebra W can be realized in terms ofthe generators E r,s of b gl ( ∞ ) after introducing the linear relation W m,n = X r ∈ Z +1 / q − ( r +1 / n E m − r,r , (2.2)and upon identification of the central elements C [13]. Through this relation, the representations3f b gl ( ∞ ) are naturally lifted to representations of quantum W ∞ . In this paper, we will consideronly two representations referred as Dirac representation and βγ -representation .As the name suggests, the Dirac representation ρ ( D ) acts on the Fock space of a 2D Diracfermion with modes ψ r , ¯ ψ r satisfying the Clifford algebra { ψ r , ¯ ψ s } = δ r + s . The Fock space is builtupon the vacuum state |∅i , annihilated by positive modes, from the action of negative modes. Therepresentation has level one and follows from the assignment ρ ( D ) ( E r,s ) =: ¯ ψ r ψ s : where :: denotesthe normal ordering with positive modes moved to the right [44]. It is deeply connected to theSchur polynomials and will be studied in great details in this section.On the other hand, the βγ -representation ρ ( βγ ) has level minus one. It acts on the Fock space ofthe ( β, γ ) bosonic ghost system with modes β r , γ r satisfying the Heisenberg algebra [ β r , γ s ] = − δ r + s [45]. The states are built in the same manner from a vacuum state |∅i annihilated by the positivemodes of both fields, β r |∅i = γ r |∅i = 0 for r >
0, and the generators are also represented in thesame way ρ ( βγ ) ( E r,s ) =: β r γ s :.Both algebras b gl ( ∞ ) and W admit a co-commutative coproduct, and representations of higherinteger level | C | > N Dirac or βγ representa-tions. Remark
In the limit q →
1, the algebra W reduces to the W ∞ algebra and the modes W m,n can be expanded in powers of ( nα ) where q = e − α . The k th power W ( k ) m corresponds to a linearcombination of the m th modes of the currents of spin lower or equal to k + 1, W m,n = ∞ X k =0 ( nα ) k k ! W ( k ) m , W ( k ) m = X r ∈ Z +1 / ( r + 1 / k E m − r,r . (2.3)Most of the results presented in this paper have their counterpart in the W ∞ algebra. In particular,the commutative subalgebra appearing in the vertical presentation coincides with the zero modes ofthe currents of spin k that are indeed commutative. For more information on the relations betweenBCD subalgebras of b gl ( ∞ ) and subalgebras of W ∞ , we refer to the very complete paper of Kac,Wang and Yan [46]. Their results may be relevant to the study of the 4 D N = 2 gauge theories inthe self-dual omega-background (i.e. C ε × C ε at β = − ε /ε = 1) [25]. We denoted ρ ( D ) the Dirac representation of the algebra W , but omit the notation if no confusionensues. The representation of the generators W m,n follows from the relation 2.2, W m,n = X r ∈ Z +1 / q − ( r +1 / n : ¯ ψ m − r ψ r : . (2.4)4ntroducing the fermionic fields ψ ( z ) = X r ∈ Z +1 / z − r − / ψ r , ¯ ψ ( z ) = X r ∈ Z +1 / z − r − / ¯ ψ r (2.5)that satisfy the anticommutation relation { ψ ( z ) , ¯ ψ ( w ) } = z − δ ( z/w ) with the multiplicative Diracdelta function δ ( z ) = P k ∈ Z z k , the generators can be realized as the contour integrals W m,n = I dz iπ z m : ¯ ψ ( z ) ψ ( q n z ) : . (2.6)The adjoint action of the algebra W on the fermionic fields reads[ W m,n , ψ ( z )] = − z m ψ ( q n z ) , [ W m,n , ¯ ψ ( z )] = q ( m − n z m ¯ ψ ( q − n z ) . (2.7)The Dirac fermion Fock space has a PBW basis spanned by the states | λ i labeled by Youngdiagrams λ = ( λ , λ , · · · ), | λ i = d ( λ ) Y i =1 h ( − λ ′ i + i ¯ ψ − ( λ i − i +1 / ψ − ( λ ′ i − i +1 / i |∅i , (2.8)where λ ′ is the transposed of the Young diagram λ , and d ( λ ) the number of boxes on the diagonal.We recognize in the indices the arm length a ( ) = λ i − j and leg length l ( ) = λ ′ j − i of the boxes= ( i, j ) ∈ λ on the diagonal (i.e. i = j ). These states are called Schur states for a reason thatwill become obvious shortly.
Bosonization
The Dirac representation can be rewritten as an action on the Fock space of a2D free boson using the celebrated bosonization technique. The bosonic Fock space is built uponthe action of the negative modes a − k of an Heisenberg algebra [ a k , a l ] = kδ k + l on the vacuum |∅i annihilated by positive modes. The bosonization formulas,¯ ψ ( z ) = ... e φ ( z ) ... , ψ ( z ) = ... e − φ ( z ) ... , ... ¯ ψ ( z ) ψ ( z )... = ∂φ ( z ) , with φ ( z ) = Q + a log z − X k ∈ Z × k z − k a k , (2.9)also involve a zero mode a and its dual operator Q such that [ a , Q ] = 1. To distinguish it fromthe fermionic normal-ordering, we denote the bosonic normal-ordering with the symbols ... · · · ..., itconsists in moving to the right the modes a k with k ≥
0. Strictly speaking, the fermionic Fockspace F decomposes into a direct sum of bosonic modules V α in which a acts as a constant α .However, the representation of the quantum W ∞ generators do not involve the operator Q , thus a is central and we can restrict ourselves to a single module V α , we take here α = 0.5he relation between bosonic and fermionic modes a k = X r ∈ Z +1 / : ¯ ψ k − r ψ r : (2.10)leads to identify the Heisenberg modes a k with the Dirac representation of the generators W k, of W . The representation for the modes W m,n with n = 0 is more easily formulated using the currents w n ( z ) = (1 − q n ) X m ∈ Z z − m W m,n + C, (2.11)for which the representation reads w n ( z ) = (1 − q n ) z ¯ ψ ( z ) ψ ( q n z ) = q − na ... exp − X k ∈ Z × z − k k (1 − q − nk ) a k ! ... . (2.12)The modules V α are isomorphic to the ring of symmetric polynomials with infinitely manyvariables. This correspondence sends the symmetric power sums p k ( x ) = P i x ki to the creationoperators a − k , and the constant polynomial 1 to the vacuum state |∅i . Moreover, it follows fromthe Frobenius formula that the Schur states | λ i defined in 2.8 are the image of the Schur polynomials s λ ( x ) [47]. They can be computed explicitly by decomposition of these polynomials in the PBWbasis of symmetric power sums, replacing p λ ( x ) k · · · p λ n ( x ) k n ↔ ( a − λ ) k · · · ( a − λ n ) k n |∅i . (2.13)It is also possible to define the adjoint basis acting with a k = ( a − k ) † and Q † = − Q , a † = a (or ¯ ψ † r = ψ − r , ψ † r = ¯ ψ − r ) on the dual vacuum h∅| . The Schur states obtained in this way areorthonormal, h λ || µ i = δ λ,µ . The quantum W ∞ algebra is a limit of the quantum toroidal gl (1) algebra, or Ding-Iohara-Mikialgebra [15, 16]. The latter is known to possess two central elements ( c , c ), and two associatedHeisenberg subalgebras [ a ( i ) k , a ( i ) l ] = c i kδ k + l (upon normalization). The Fock representation haslevels (1 , presentations of theFock representation.In the horizontal presentation , the focus is on the Heisenberg algebra a (1) k withrespect to which the generators are expressed as vertex operators [48]. On the other hand, inthe vertical presentation , the focus is on the commuting operators a (2) k that act diagonally on a6hosen basis [49]. These two dual pictures are exchanged by the action of Miki’s automorphismthat switches the two subalgebras and maps ( c , c ) → ( − c , c ) [16]. The quantum W ∞ algebra inherits of the same properties. In the definition 1.1, the secondcentral charge c has already been set to zero, while we can identify c = C [17]. The two subalgebrascorrespond to a (1) k = W k, and a (2) k = W ,k . The horizontal presentation can be read from theequation 2.12 that indeed expresses the currents w n ( z ) built upon the generators W m,n as vertexoperators in the modes a (1) k = a k . In this subsection, we derive the vertical presentation in which theaction of a (2) k = W ,k is diagonal. Roughly speaking, Miki’s automorphism sends W m,n to W − n,m ,thereby exchanging the two subalgebras [17, 37].It is not difficult to derive the vertical presentation from the fermionic formulas for the states2.8 and the generators 2.4. However, in order to write down this action in a form similar to thevertical presentation of the quantum toroidal gl (1) algebra in [31, 36], it is necessary to introduceseveral notations. Following ref. [8], we call a border strip of length m a set of m adjacent boxes(i.e. sharing a single edge). We denote A m ( λ ) (resp. R m ( λ )) the set of border strips of lengths m that can be added to (resp. removed from) the Young diagram λ such that λ ± ρ is a valid Youngdiagram. In addition, to any box of a Young diagram, we assign a complex number called the boxcontent , and obtained as χ = q i − j ∈ C × where ( i, j ) are the coordinates of the box. For each strip ρ ∈ A m ( λ ) ∪ R m ( λ ), we associate the complex number χ ρ corresponding to the content of the topleft box. Thus, the contents of the boxes in the strip ρ are given by q i − χ ρ for i = 1 · · · m . Finally,we denote r ( ρ ) (resp. r ( ρ ′ )) the number of rows (columns) that a strip ρ occupies minus one, wehave r ( ρ ) + r ( ρ ′ ) = m − m > W ,n | λ i = − (1 − q − n ) X ∈ λ χ n | λ i ,W m,n | λ i = q − n X ρ ∈ R m ( λ ) ( − r ( ρ ′ ) χ nρ | λ − ρ i ,W − m,n | λ i = q ( m − n X ρ ∈ A m ( λ ) ( − r ( ρ ′ ) χ nρ | λ + ρ i , (2.14)and, in particular, a k | λ i = X ρ ∈ R k ( λ ) ( − r ( ρ ′ ) | λ − ρ i , a − k | λ i = X ρ ∈ A k ( λ ) ( − r ( ρ ′ ) | λ + ρ i . (2.15) We insist here on the term presentation because in this picture the representation ρ is fixed, we simply expressthe action of the generators differently: the horizontal presentation refers to the expression of the Drinfeld currents x ± ( x ), ψ ± ( z ) as vertex operators while the vertical presentation refers to the action of the dual currents S · x ± ( z ), S · ψ ± ( z ) on the Fock states ( S denoting Miki’s automorphism). Note that the profile of the Young diagram meet the diagonals i − j =fixed at only one point. As a result, thecontent χ ρ characterizes uniquely the strips ρ ∈ A m ( λ ) ∪ R m ( λ ) at fixed m . d such that [ d, W m,n ] = − mW m,n . It isrepresented as d = X r ∈ Z +1 / r : ¯ ψ − r ψ r : ⇒ d | λ i = | λ | | λ i , α d W m,n α − d = α − m W m,n . (2.16) The correspondence between the Schur states | λ i of the bosonic Fock space and the Schur symmetricpolynomials s λ ( x ) provides an expression for the Schur polynomials as bosonic correlators, s λ ( x ) = h∅| e P k> pk ( x ) k a k | λ i . (2.17)In practice, the polynomial is obtained by expanding the exponential as an infinite series. Since a k acts on the Schur states by removing strips of boxes (following equ. 2.15), the r.h.s. contains onlya finite number of terms. The proof of this formula is not easily found in the litterature and wedecided a short derivation in the appendix A.As we shall see, the formula 2.17 for the Schur polynomials, combined with the expression 2.15for the vertical action of the generators W m,n , can be used to recover several important propertiesof the Schur polynomials. As a warm-up, we can introduce the exponentiated grading operator α d on the left of the correlator since h∅| α d = h∅| . Commuting the operator all the way to the rightusing the formulas 2.16, we recover the identity s λ ( αx ) = α | λ | s λ ( x ) expressing the fact that Schurpolynomials are homogeneous of degree | λ | .Recursion relations, reminiscent of the Pieri rules, are obtained in a similar manner from the in-sertion of the operator a − k inside the correlator 2.17. Commuting the operator with the exponential,we find 0 = h∅| a − k e P k> pk ( x ) k a k | λ i = − p k ( x ) s λ ( x ) + h∅| e P k> pk ( x ) k a k a − k | λ i . (2.18)Evaluating the action of a − k on the Schur states using the vertical presentation 2.15, we recoverindeed the rules (see [8], page 48) p k ( x ) s λ ( x ) = X ρ ∈ A k ( λ ) ( − r ( ρ ′ ) s λ + ρ ( x ) . (2.19)The usual Pieri rules for Schur polynomials involve either the complete symmetric functions h k orthe elementary symmetric function e k . Here, instead, we multiply by the elementary power sums p k , which explains the presence of the extra signs in the r.h.s..This derivation can be easily generalized to skew-Schur polynomials using the formula A.1 ofthe appendix and the contragredient action of a − k , we find p k ( x ) s λ/µ ( x ) = X ρ ∈ A k ( λ ) ( − r ( ρ ′ ) s ( λ + ρ ) /µ ( x ) − X ρ ∈ R k ( µ ) ( − r ( ρ ′ ) s λ/ ( µ − ρ ) ( x ) . (2.20)8urthermore, taking the derivative ∂/∂p k on both sides of the equation 2.17, and using the expression2.15 for the action of a k on Schur states, we find k ∂∂p k s λ ( x ) = X ρ ∈ R k ( λ ) ( − r ( ρ ′ ) s λ − ρ ( x ) . (2.21)The algebra W is multiplicatively generated by the modes W m, and W ,n , so it is sufficient toexamine the action of the remaining modes W ,n . As shown in appendix B, the insertion of thesemodes in the correlator 2.17 generates a q-difference equation solved by the Schur polynomials,namely N X i =1 Y j = i x i − q n x j x i − x j T q − n ,x i s λ ( x ) = N X i =1 q n ( i − + q − n (1 − q n ) X ∈ λ χ n ! s λ ( x ) , (2.22)where, following [8], we denoted T q − n ,x i the operation consisting in replacing the variable x i by q − n x i . The number N > ℓ ( λ ) of variables x i has been re-introduced to play the role of a cut-off.In this equation, the dependence in n arises only through the variable q n , and thus we have onlya single q-difference equation. In fact, the l.h.s. can be identified with the Macdonald operator t − N D N after the replacement ( t, q ) → ( q − n , q − n ) (recall that Schur polynomials are a specializationof Macdonald polynomials at t = q ). The dependence in the number N of variables drops if wedefine E = q n N X i =1 Y j = i x i − q n x j x i − x j T q − n ,x i − N X i =1 q ni ⇒ Es λ ( x ) = (1 − q n ) X ∈ λ χ n s λ ( x ) . (2.23)This expression is suitable for taking the limit N → ∞ . Macdonald polynomials
The presence of the Heisenberg subalgebra formed by the modes W k, is essential for the derivation of these formulas. The quantum toroidal gl (1) algebra also con-tains a Heisenberg subalgebra and the same considerations apply to Macdonald polynomials. Thederivation of the appendix A extends to this case, and provides the formula P λ ( x ) = h∅| e P k> pk ( x ) k − tk − qk a k | P λ i , (2.24)in terms of the Macdonald states | P λ i [36]. Then, the combination of horizontal and vertical actionsproduce both the Pieri rules and the Macdonald operators [48]. In the next sections, we will attemptto apply this generic program to the subalgebras ˆ B ∞ , ˆ C ∞ and ˆ D ∞ . Hamiltonians
We would like to conclude this section with an important remark. Since theSchur polynomials do not depend on the parameter q n , the q-difference equation 2.22 produces in9he limit q → q n = e − α , we can derive easily the first two differential operators in the limit α → N X i =1 Y j = i q − n x i − x j x i − x j T q − n ,x i = N + α H + α H + O ( α ) . (2.26)The Hamiltonian H k gives the action of the zero mode of the current of spin k for the W ∞ algebrafound in the degenerate limit q →
1. After a short calculation, we find the expressions H = X i D i + 12 N ( N − , H = 12 H β =1CS + 12 ( N − H − N ( N − N − , (2.27)where we denoted D i = x i ∂ x i and H β =1CS is the Calogero-Sutherland Hamiltonian specialized at β = 1, H β CS = X i ( x i ∂ x i ) + β X i,ji It is equivalent, and sometimes more convenient, to present the algebra ˆ B ∞ using gen-erators with integer indices O ( B ′ ) i,j = O ( B ) i +1 / ,j − / . These generators obey the commutation relations[ O ( B ′ ) i,j , O ( B ′ ) k,l ] = δ j + k O ( B ′ ) i,l − δ i + l O ( B ′ ) k,j + 2 C ( θ ( i + 1 / − θ ( k + 1 / δ j + k δ i + l + ( − k + l h δ i + k O ( B ′ ) l,j − δ j + l O ( B ) i,k + 2 C ( θ ( l + 1 / − θ ( i + 1 / δ i + k δ j + l i , (3.6)together with the property O ( B ′ ) i,j + ( − i + j O ( B ) j,i = − Cδ i, δ j, . The algebra B ∞ is recovered usingthe representation of level zero of b gl ( ∞ ) ρ ′ ( E r,s ) = e r − / , − s − / (it is a shift of the representation ρ defined in the previous section). From 3.2, we deduce that ρ ′ ( O ( B ′ ) i, − j ) = b i,j with b i,j = e i,j +( − i + j +1 e − j, − i . 11 .1 Representation on the Majorana fermion Fock space The algebra ˆ B ∞ can be represented on the Fock space of a self-adjoint fermion, also called neutralor Majorana fermion, with periodic (or Ramond) boundary conditions ˜ φ ( e iπ z ) = ˜ φ ( z ) [4]. Thisfield decomposes on integer modes ˜ φ ( z ) = X k ∈ Z z − k ˜ φ k , { ˜ φ k , ˜ φ l } = ( − k δ k + l ⇒ { ˜ φ ( z ) , ˜ φ ( w ) } = δ ( − z/w ) . (3.7)The modes ˜ φ k square to zero, except for the zero mode ( ˜ φ ) = 1 / 2. The Fock space is obtainedfrom the action of the modes ˜ φ − k for k ≥ |∅i B annihilated by the strictly positivemodes, i.e. φ k |∅i B = 0 for k > 0. The normal ordering is usually defined by moving the positivemodes to the right, but here we need to take into account the zero mode, and define instead: ˜ φ k ˜ φ l := ˜ φ k ˜ φ l − B h∅| ˜ φ k ˜ φ l |∅i B , : ˜ φ ( z ) ˜ φ ( w ) := ˜ φ ( z ) ˜ φ ( w ) − B h∅| ˜ φ ( z ) ˜ φ ( w ) |∅i B , with B h∅| ˜ φ k ˜ φ l |∅i B = ( − k δ k + l θ ( k ) , B h∅| ˜ φ ( z ) ˜ φ ( w ) |∅i B = 12 z − wz + w , (3.8)where the Heaviside function takes the value θ ( k ) = 1 / k = 0 (note that : ˜ φ k ˜ φ l := − : ˜ φ l ˜ φ k ).The Majorana representation has level 1 / 2, it is defined as ρ ( ˜ M ) ( O ( B ′ ) i,j ) = ( − j : ˜ φ i ˜ φ j : − δ i, δ j, ⇒ ρ ( ˜ M ) ( W Bm,n ) = X k ∈ Z ( − k q − nk : ˜ φ m − k ˜ φ k : − δ m, . (3.9)We denote this representation ρ ( ˜ M ) but omit the notation if no confusion ensues. It is sometimesuseful to express the generators W Bm,n as a contour integral of the Majorana fermionic field, W Bm,n = I dz iπ z m − : ˜ φ ( z ) ˜ φ ( − q n z ) : − δ m, . (3.10)The modes act on the fermionic field as[ W Bm,n , ˜ φ ( z )] = q − mn z m ˜ φ ( q − n z ) + ( − m +1 z m ˜ φ ( q n z ) , (3.11)and, in particular, [ W Bm, , ˜ φ ( z )] = 2 δ m, odd z m ˜ φ ( z ) where δ m, odd is one if m is even and zero otherwise. Decomposition of the Dirac representation As a subalgebra of W , the algebra W B alsopossesses a Dirac representation. From the action 3.4 of the automorphism σ B on the contourintegral expression 2.6 of the generators W m,n , we find for the subalgebra generators the formula ρ ( D ) ( W Bm,n ) = I dz iπ z m (cid:0) : ¯ ψ ( z ) ψ ( q n z ) : + q n : ¯ ψ ( − q n z ) ψ ( − z ) : (cid:1) − δ m, . (3.12) Instead, the Majorana fermion with anti-periodic (or Neuveu-Schwarz) boundary conditions, φ ( e iπ z ) = − φ ( z ),which is decomposed on half-integer modes, will appear in the representation of the subalgebra W D below (note,however, that the field will be multiplied by an extra power z / to get rid of the sign). 12t is well known that the charged Dirac fermion can be decomposed into two neutral Majoranafermions, ψ ( z ) = 1 √ (cid:16) ˜ φ ( − z ) + i ˜ φ ( − z ) (cid:17) , ¯ ψ ( z ) = z − √ (cid:16) ˜ φ ( z ) − i ˜ φ ( z ) (cid:17) . (3.13)Plugging this decomposition into the previous expression of the generators W Bm,n , we find ρ ( D ) ( W Bm,n ) = I dz iπ z m − (cid:16) : ˜ φ ( z ) ˜ φ ( − q n z ) : + : ˜ φ ( z ) ˜ φ ( − q n z ) : (cid:17) − δ m, , (3.14)which corresponds to the fact that the Dirac representation of W B is not irreducible but decomposesinto two Majorana representations as ρ ( D ) = ρ ( ˜ M ) ⊗ ⊗ ρ ( ˜ M ) . Bosonization The Majorana fermion can be bosonized using a Heisenberg algebra a k with onlyodd modes k , and such that [ a k , a l ] = 2 kδ k + l (the extra factor 2 in the r.h.s. is introduced forlater convenience). In the mathematics literature, this is known as the twisted boson-fermioncorrespondence [50], but it is also sometimes simply referred as the bosonization of type B. TheMajorana field is realized as˜ φ ( z ) = 1 √ e − P k ∈ Z δ k, odd z − kk a k ... ⇒ ˜ φ ( z ) ˜ φ ( w ) = z − wz + w ... ˜ φ ( z ) ˜ φ ( w )... ., (3.15)where we denoted again ... · · · ... the bosonic normal ordering. Note that the adjoint action a † k = a − k corresponds to ˜ φ ( z ) † = ˜ φ ( − z − ) in the case of Majorana, and ψ ( z ) † = z − ¯ ψ ( z − ) in the case ofDirac. Computing W Bk, in the Majorana representation 3.9 leads to identify the Heisenberg algebras a k = W Bk, . The expression for the other generators is found by performing the bosonization on thecontour integral formula 3.10, W Bm,n = 12 1 + q n − q n I dz iπ z m − ... e − P k ∈ Z δ k, odd z − kk (1 − q − nk ) a k ... − δ m, − q n . (3.16)In this way, we recover the vertex representation for the algebra W B proposed in [13]. The Fock space for the periodic Majorana fermion is spanned by the states | λ i B labeled by thesymmetric partitions λ = λ ′ . These states are defined using the hook decomposition of the Youngdiagram, assigning the action of the modes ˜ φ − k i to the i th hook of length h i = 2 k i + 1, | λ i B = ˜ φ − k · · · ˜ φ − k d ( λ ) |∅i B , k i ∈ Z . (3.17)The hooks are strictly ordered, and thus k > k > · · · > k d ( λ ) . In this setting, the action of thezero mode ˜ φ is to add/remove a box on the diagonal (with a multiplication by a factor 1 / Alternatively, following a well-known bijection, the set of hooks can be seen as defining a strict partition. φ †− k = ( − k ˜ φ k (with k > 0) on the dual vacuum B h∅| , they are orthonormal: B h λ || µ i B = δ λ,µ .In order to write down the vertical presentation, we need to introduce again several notations.To keep the Young diagram symmetric, the horizontal strips will be added/removed in pairs. Forthis purpose, we define the sets A Bm ( λ ) (resp. R Bm ( λ )) of strips with m -boxes lying strictly belowthe diagonal that can be added to (resp. removed from) the symmetric partition λ . Such strips ρ have a unique transposed ρ ′ ∈ A m ( λ ) (resp. ρ ′ ∈ R m ( λ )) lying strictly above the diagonal. Asbefore, the content χ ρ of a strip ρ ∈ A Bm ( λ ) ∪ R Bm ( λ ) is defined as the content of its box in the topleft corner. Thus, the transposed strip has the content χ ρ ′ = q − m χ − ρ . The modes W Bm,n act byadding or removing the pairs ( ρ, ρ ′ ) and we denote the corresponding state | λ ± ρ i B . We keep thenotation r ( ρ ) for the number of rows that the strip ρ occupies minus one.In addition, the vertical action also involves the addition/removal of a pair of hooks. We denote A HPm ( λ ) (resp. R HPm ( λ )) the set of hook pairs with a total of 2 m boxes, that can be added to (resp.removed from) the symmetric diagram λ . These hooks are not necessarily adjacent, and they canbe inserted/removed anywhere in the diagram. For ρ ∈ A HPm ( λ ) ∪ R HPm ( λ ), we denote | λ ± ρ i B thestate corresponding to the addition/removal of the hook pair ρ . For a hook pair ρ , let k ( ρ ) be theinteger such that the smaller hook has 2 k ( ρ ) + 1 boxes. The content is defined as the box contentof the box in the top left corner of the longest hook, and thus χ ρ = q k ( ρ ) − m +1 . Finally, we denote r ( ρ ) + 1 the number of hooks of the diagram λ lying between the two hooks to be inserted/removed(we omit the dependence in the partition λ for this notation).The action of the operators W Bm,n on the states | λ i B is obtained by a direct, but tedious, calcu-lation, W B ,n | λ i B = (1 − q n ) X ∈ λ − ( q − n χ n + χ − n ) − | λ i B ,W Bm,n | λ i B = X ρ ∈ R Bm ( λ ) (cid:0) q − ( m − n χ − nρ + ( − m +1 q − n χ nρ ) (cid:1) ( − r ( ρ ) | λ − ρ i B , + X ρ ∈ R HPm +1 ( λ ) (cid:18) − δ k ( ρ ) , (cid:19) (cid:0) q − mn χ − nρ + ( − m +1 χ nρ (cid:1) ( − r ( ρ )+ k ( ρ )+ m | λ − ρ i B ,W B − m,n | λ i B = X ρ ∈ A Bm ( λ ) (cid:18) − δ χ ρ ,q (cid:19) (cid:0) q n χ − nρ + ( − m +1 q ( m − n χ nρ (cid:1) ( − r ( ρ ) | λ + ρ i B , + X ρ ∈ A HPm +1 ( λ ) (cid:0) q mn χ nρ + ( − m +1 χ − nρ (cid:1) ( − r ( ρ )+ k ( ρ )+1 | λ + ρ i B . (3.18)In the first line, the sum is performed on the set of boxes λ − lying strictly below the diagonal.The Kronecker deltas provide an extra factor 1 / φ is involved. These rules might appearcomplicated, but they can be easily implemented on a computer program. In particular, we have14or a k = W Bk, with k odd: a k | λ i B = 2 X ρ ∈ R Bk ( λ ) ( − r ( ρ ) | λ − ρ i B + 2 X ρ ∈ R HPk +1 ( λ ) (cid:18) − δ k ( ρ ) , (cid:19) ( − r ( ρ )+ k ( ρ )+ k | λ − ρ i B ,a − k | λ i B = 2 X ρ ∈ A Bk ( λ ) (cid:18) − δ χ ρ ,q (cid:19) ( − r ( ρ ) | λ + ρ i B + 2 X ρ ∈ A HPk +1 ( λ ) ( − r ( ρ )+ k ( ρ )+1 | λ + ρ i B . (3.19)It might be surprising to notice that the action of the operators W Bm,n produces states with adifferent number of boxes: the addition/removal of two strips of length m changes the number ofboxes by 2 m while the addition/removal of a pair of hooks modifies it by 2 m + 2. However, in bothcases, the number of boxes strictly below the diagonal | λ − | varies by ± m . This property can beformally encoded in the action of a grading operator d B such that [ d B , W Bm,n ] = − mW Bm,n , d B = 12 X k ∈ Z ( − k k : ˜ φ − k ˜ φ k : ⇒ [ d B , ˜ φ k ] = − k ˜ φ k , d B | λ i B = | λ − | | λ i B , α d B W Bm,n α − d B = α − m . (3.20)Furthermore, the number of hooks varies by 0 , ± W Bm,n . Thus,the Majorana representation is not irreducible, but it decomposes into an action on symmetricpartitions with either an even or an odd number of hooks. This is the reason for introducing twosets of symmetric polynomials b λ ( x ) and b ∗ λ ( x ) below. The bosonic formula 2.17 for the Schur polynomials can be generalized to the Majorana fermionand its bosonization, it defines two sets of polynomials labeled by symmetric partitions, b λ ( x ) = B h∅| e P k> δ k, odd pk ( x ) k a k | λ i B , b ∗ λ ( x ) = B h | e P k> δ k, odd pk ( x ) k a k | λ i B . (3.21)The action of the modes a k on the states | λ i B is given by the vertical formula 3.19, it produces onlystates labeled by partitions with a strictly smaller number of boxes, and so the expansion of ther.h.s. is finite. We provide here a few examples: b ∅ = 1 , b = p , b = p , b = 23 p − p , b = 23 p + 13 p , b = 23 p − p p ,b ∗ = 12 , b ∗ = p , b ∗ = p , b ∗ = 13 p − p , b ∗ = 23 p + 13 p , b = 13 p + 23 p p . (3.22) Since λ is symmetric, | λ | = 2 | λ − | + d ( λ ) and the parity of the number of hooks is the same as the parity of thenumber of boxes. q , and that they involve onlyodd power sums by definition. Inserting the grading operator in the correlator, one can show, likein the Schur case, that they are homogeneous polynomials of degree | λ − | . Moreover, it is easily seenfrom the action of a k that b λ ( x ) is non-zero for | λ | even and b ∗ λ ( x ) for | λ | odd. In fact, they can beseen as special cases of the more general skew polynomials b λ/µ ( x ) = B h µ | e P k> δ k, odd pk ( x ) k a k | λ i B . (3.23)From the action 3.19 of a k , we observe that this expression is non-vanishing only if the Youngdiagram µ is contained in λ .In the following, we focus mostly on the polynomial b λ ( x ). This polynomials is, up to normal-ization, a Q-Schur function Q s ( λ ) = 2 ♯ b λ for the strict partition s ( λ ) uniquely associated to thesymmetric partition λ and defined by the hook lengths, s ( λ ) = (( h − / , ( h − / , · · · ). Thisidentification follows from the fermionic expression, b ( N ) λ ( x ) = 2 N/ B h∅| ... ˜ φ ( − x − ) ˜ φ ( − x − ) · · · ˜ φ ( − x − N )... | λ i B = 2 N/ N Y i,j =1 i Like the Schur polynomials, the polynomials b λ ( x ) satisfy a q-difference equationand are independent of q . Expanding at q = 1, it is possible to derive a tower of differentialequations satisfied by these polynomials. However, since the q-difference equation is antisymmetricunder the exchange q → q − , only odd powers of α remain in the expansion of q n = e − α . Theexpansion easily follows from the following fact, T B = X i (cid:16) B ( n ) i ( x ) T q − n ,x i − B ( − n ) i ( x ) T q n ,x i (cid:17) = ∆ B ( x ) − X i ( T q − n ,x i − T q n ,x i ) ∆ B ( x ) , with ∆ B ( x ) = N Y i,j =1 i We review here the bosonization performed by Van de Leur, Orlov andShiota in [10] (following a suggestion from Date, Jimbo, Kashiwara and Miwa [3]). It is sometimescalled the twisted bosonization , and describes, in fact, a correspondence between the symplecticboson and a boson+fermion system. The bosonic modes correspond to the modes W Cm,n with n = 0 and m odd that form the Heisenberg subalgebra. In this section, we denote a k = W Ck, forshort, we have [ a k , a l ] = − kδ k + l for this representation of level C = − / 2. Following [10], weintroduce a dressed version of the bosonic field, χ ( z ) = V − ( z ) ˆ φ ( z ) V + ( z ) , V ± ( z ) = e ∓ P k> δ k, odd z ∓ kk a ± k . (4.15)The dressing is chosen such that the field χ ( z ) commutes with the modes a k . From the commutator[ a k , ˆ φ ( z )] = 2 z k ˆ φ ( z ), we deduce the exchange relations V ± ( z ) ˆ φ ( w ) = ± (cid:18) z − wz + w (cid:19) ± ˆ φ ( w ) V ± ( z ) , V + ( z ) V − ( w ) = z + wz − w V − ( w ) V + ( z ) , (4.16)which leads to χ ( z ) χ ( w ) = z − wz + w V − ( z ) V − ( w ) ˆ φ ( z ) ˆ φ ( w ) V + ( z ) V + ( w ) . (4.17)After a short computation that we chose not to reproduce here as it is found in [10], it is possibleto show that the field χ ( z ) anticommutes with itself, and thus should be thought of as a fermionic It differs from the untwisted bosonization introduced by Anguelova in [9, 55]. χ ( z ) = X r ∈ Z +1 / z − r − / χ r , { χ r , χ s } = 2 r ( − r − / δ r + s . (4.20)The symplectic boson vacuum state is annihilated by both bosonic and fermionic positive modes, a k |∅i C = χ r |∅i C = 0 for k, r > 0. For the bosonic modes, it follows from the vertical presentationgiven below, while for the fermionic modes, it is seen using the definition 4.15 after expanding inpowers of z . It leads to define a fermionic normal-ordering by moving positive modes to the right.Then : χ ( z ) χ ( w ) := χ ( z ) χ ( w ) − C h∅| χ ( z ) χ ( w ) |∅i C , C h∅| χ ( z ) χ ( w ) |∅i C = z − w ( z + w ) . (4.21)Thus, the horizontal presentation of the representation involves not only the bosonic modes a k ,but also the fermionic ones χ r . As we shall see, this fact renders the analysis of the correspondingsymmetric polynomials much more complicated. The horizontal presentation can be written interms of the currents w Cn ( z ) = (1 − q n ) X m ∈ Z z − m W Cm,n + 2 C. (4.22)In the symplectic boson representation, the currents reads w Cn ( z ) = − (1 − q n ) z ˆ φ ( z ) ˆ φ ( − q n z ). Afterthe twisted bosonization, they become w Cn ( z ) = − (1 − q n ) q n ze − P k> δ k, odd (1 − q nk ) zkk a − k χ ( z ) χ ( − q n z ) e P k> δ k, odd (1 − q − nk ) z − kk a k . (4.23) There are several ways to label the PBW basis of the symplectic boson Fock space. Here, we followagain the reference [10] and use the set of partitions with odd parts, denoted OP, and representedas Young diagrams with an odd number of boxes in each column. The basis consists of the vectors | λ i C = ˆ φ − λ / · · · ˆ φ − λ ℓ / |∅i C , λ ∈ OP , (4.24) The expression of the commutator of the field is more involved, it read { χ ( z ) , χ ( w ) } = z − ∆( z/w ), where weintroduced the distribution ∆( z ) = δ ( − z ) − δ ′ ( − z ) with δ ′ ( z ) = X k ∈ Z kz k = z∂ z δ ( z ) ⇒ f ( z ) δ ′ ( z/a ) = f ( a ) δ ′ ( z/a ) − af ′ ( a ) δ ( z/a ) . (4.18)In fact, the distribution ∆( z ) can also be obtained from the difference of the expansions in powers of z ± of thevacuum expectation values C h∅| χ ( z ) χ ( w ) |∅i C , (cid:20) z − w ( z + w ) (cid:21) + − (cid:20) z − w ( z + w ) (cid:21) − = z − ∆( z/w ) . (4.19) C h λ | is obtained from the action of the adjoint modes ˆ φ †− r = ( − r +1 / ˆ φ r with r > C h∅| .In order to formulate the vertical presentation of the action of the generators W Cm,n on thisbasis, we need to introduce again several new notations. Firstly, to a box ∈ λ of coordinates( i, j ), we associate a new box content ˆ χ = q ( j − / depending only on the second coordinate j .Next, we introduce the notion of disjoint vertical strips that are sets of columns of boxes, the boxescoordinates ( i k , j k ) satisfying i k +1 ≥ i k and j k +1 = j k − A Cm ( λ ) (resp. R Cm ( λ )) the set of disjoint strips ρ with 2 m boxes that can be added to (resp. removed from) λ ,such that λ ± ρ ∈ OP and without changing the number of columns (i.e. ℓ ( λ ± ρ ) = ℓ ( λ )). Tosuch a strip we associate the content ˆ χ ρ = ˆ χ where is the box in the top right corner (i.e. ofmaximal coordinate j ). Thus, the boxes of a disjoint vertical strip have the contents q − k/ ˆ χ ρ for k = 0 , · · · m − A CPm ( λ ) (resp R CPm ( λ )) of columns pairs with a total of 2 m boxes that can be added to (resp. removed from) λ . To a pair of columns ρ ∈ A CPm ( λ ) ∪ R CPm ( λ ),we associate the content ˆ χ ρ = ˆ χ where is the box at the top of the highest column (i.e. again ofmaximal coordinate j ). We also associate the sign ( − ρ = ( − ( k ( ρ ) − / where k ( ρ ) is the heightof the smallest column (the tallest column has the height 2 m − k ( ρ )). In fact A CPm ( λ ) contains all pairs of column of odd heights with 2 m boxes while R CPm ( λ ) is the set of the pairsof columns of λ containing 2 m boxes. Note also that we need to count multiplicities: if λ contains k i columns withthe same height λ i , and k j columns of height λ j with λ i + λ j = 2 m , R CPm ( λ ) contains k i k j pairs of columns ( λ i , λ j ),or k i ( k i − / λ i = λ j . The set R Cm ( λ ) is also degenerate: a strip has the multiplicity Q i k i if columns λ i towhich the boxes are removed have multiplicity k i . m strictly positive W C ,n | λ i C = (1 − q n/ ) q − n/ X ∈ λ ( ˆ χ n + q − n/ ˆ χ − n ) ! | λ i C ,W Cm,n | λ i C = X ρ ∈ R Cm ( λ ) (cid:0) ( − m +1 q − mn ˆ χ nρ + q − n ˆ χ − nρ (cid:1) | λ − ρ i C + X ρ ∈ R CPm ( λ ) ( − ρ (cid:0) ( − m +1 q − mn ˆ χ nρ + q − n ˆ χ − nρ (cid:1) | λ − ρ i C ,W C − m,n | λ i C = X ρ ∈ A Cm ( λ ) (cid:0) ( − m +1 ˆ χ nρ + q ( m − n ˆ χ − nρ (cid:1) | λ + ρ i C − X ρ ∈ A CPm ( λ ) (cid:18) − δ k ( ρ ) ,m (cid:19) ( − ρ (cid:0) ( − m +1 ˆ χ nρ + q ( m − n ˆ χ − nρ (cid:1) | λ + ρ i C . (4.25)The Kronecker δ provides an extra factor 1 / k ( ρ ) = m .In particular, we have for the Heisenberg subalgebra with k odd positive, a k | λ i C = 2 X ρ ∈ R Ck ( λ ) | λ − ρ i C + 2 X ρ ∈ R CPk ( λ ) ( − ρ | λ − ρ i C ,a − k | λ i C = 2 X ρ ∈ A Ck ( λ ) | λ + ρ i C − X ρ ∈ A CPk ( λ ) (cid:18) − δ k ( ρ ) ,k (cid:19) ( − ρ | λ + ρ i C . (4.26)The algebra W C can be supplemented with a grading operator d C such that [ d C , W Cm,n ] = − mW Cm,n . It is represented as d C = X r ∈ Z +1 / ( − r − / r : ˆ φ − r ˆ φ r : , d C | λ i C = | λ | | λ i C , α d C W Cm,n α − d C = α − m . (4.27)Since the number of columns varies by 0 , ± W Cm,n , this repre-sentation is not irreducible but contains two sectors corresponding to partitions with either an oddor an even number of parts. In the following, we focus on the case of ℓ ( λ ) even. In the case of ˆ C ∞ , there are two ways of defining symmetric polynomials, depending on whetherwe generalize the bosonic expression 3.21, or the field expression 3.24 of ˆ B ∞ . In the first scenario,the fermionic component of the twisted bosonization is missing and, as a result, the q-differenceequation cannot be established. The polynomials are those obtained by van de Leur, Orlov andShiota in [10] when they set their odd times to zero, and their half-integer times to p k ( x ) /k .In [10], the way around the problem of the missing fermionic component is the introduction ofa supersymmetric vertex operator with both odd integer and half-integer times. Here, we will take23 simpler approach that roughly corresponds to consider only a specific component of the super-symmetric amplitudes (up to the important question of normalization). This scenario correspondsto generalize the formula 3.24 for the polynomials b ( N ) λ ( x ) expressed as the correlator of a finitenumber of fields N (the polynomial variables x i enter as the insertion positions). In this case,the q-difference equation associated to the diagonal action of W C ,n can be derived. However, incontrast with the case of ˆ B ∞ , the inductive limit that sends the number of variables N to infinityis non-trivial here, and we were unable to define it properly. It would be interesting to derive aq-difference equation on the supersymmetric amplitudes that admit a proper large N limit but itis beyond the scope of this paper. Following van de Leur, Orlov and Shiota [10], we introduce the set of symmetric polynomials, C λ ( x ) = 12 C h∅| e P k> δ k, odd pk ( x ) k a k | λ i C . (4.28)The normalization has been modified here, it is such that C = p . Then, e.g. C = p , C = 6 p , C = 23 p + 13 p , C = 60 p , C = 43 p − p ,C = 13 p + 23 p p , C = 203 p − p p , C = 840 p . (4.29)Since the number of columns of the Young diagram λ varies by 0 , ± W Ck, on thestate | λ i C , the polynomials C λ ( x ) are non-zero only if ℓ ( λ ) (or | λ | since λ ∈ OP) is even. However,more general (skew-type) polynomials can be introduced with C λ/µ ( x ) = 12 C h µ | e P k> δ k, odd pk ( x ) k a k | λ i C . (4.30)These polynomials are non-zero when µ is a sub-diagram of λ (i.e. µ ≺ λ ) with ℓ ( λ ) − ℓ ( µ ) even.Inserting the grading operator d C , one can show that these polynomials are homogeneous of degree( | λ | − | µ | ) / ∈ Z . An explicit expression has been found for C λ ( x ) in [10] as an Hafnian of Schurfunctions associated to the Hooks ( λ i | λ j ), C λ ( x ) = Hf( s ( λ i | λ j ) ( x )). The analogue of the Cauchyidentity has also been obtained there, X λ ∈ OP C λ ( x ) C λ ( y ) = Y i,j x i y j − x i y j . (4.31)The derivation of Pieri-like rules follows from the same steps as before, namely the insertion of W Ck, inside the correlator on the left, and the use of the Heisenberg algebra relations to move it to24he right. We find for k odd : p k ( x ) C λ ( x ) = − X ρ ∈ A Ck ( λ ) C λ + ρ ( x ) + X ρ ∈ A CPk ( λ ) (cid:18) − δ k ( ρ ) ,k ) (cid:19) ( − ρ C λ + ρ ( x ) ,k ∂∂p k ( x ) C λ ( x ) = 2 X ρ ∈ R Ck ( λ ) C λ − ρ ( x ) + 2 X ρ ∈ R CPk ( λ ) ( − ρ C λ − ρ ( x ) . (4.32)On the other hand, as anticipated, we were not able to derive a q-difference equation due to themissing fermionic operators in the correlator. Instead of generalizing the bosonic formula, we can try to generalize the expression 3.24 of thesymmetric polynomials b λ ( x ) involving the Majorana fields. For this purpose, we need to considera finite number of variables N . Moreover, in the absence of a bosonic normal-ordering, spoiled bythe fermionic modes χ r , we need to introduce the notation... ˆ φ ( z ) · · · ˆ φ ( z N )... = N Y i,j =1 i 1. ˆ φ ( z )... ˆ φ ( z ) · · · ˆ φ ( z N )... = hQ Ni =1 1 z + z i i + ... ˆ φ ( z ) ˆ φ ( z ) · · · ˆ φ ( z N )...2. ... ˆ φ ( z ) · · · ˆ φ ( z N )... ˆ φ ( z ) = hQ Ni =1 1 z + z i i − ... ˆ φ ( z ) ˆ φ ( z ) · · · ˆ φ ( z N )...3. lim z →− z i ... ˆ φ ( z ) ˆ φ ( z ) · · · ˆ φ ( z N )... = Q N j =1 j = i ( z j − z i ) ... ˆ φ ( z ) · · · ✟✟✟ ˆ φ ( z i ) · · · ˆ φ ( z N )... , 4. [ ˆ φ ( z ) , ... ˆ φ ( z ) · · · ˆ φ ( z N )...] = P Ni =1 z − δ ( − z/z i ) Q N j =1 j = i ( z i + z j ) ... ˆ φ ( z ) · · · ✟✟✟ ˆ φ ( z i ) · · · ˆ φ ( z N )... In order to obtain the expansion in powers of z ∓ , we assumed that the operators are evaluated between twostates C h λ | · · · | µ i C . Then, the insertion of the field ˆ φ ( z ) on the left will produce only a finite number of positivepowers, while an insertion on the right produces a finite number of negative powers. The third property follows fromthe normal ordering ( z + w ) ˆ φ ( z ) ˆ φ ( w ) = 1 + ( z + w ) : ˆ φ ( z ) ˆ φ ( w ) :. Each matrix element C h λ | : ˆ φ ( z ) ˆ φ ( w ) : | µ i C has only a finite number of terms of order z k w l , and so the normal-ordered product is non-singular at z + w = 0.Thus, the limit z → − w of the r.h.s. is simply one. Taking the difference of the first two properties, a delta functionappears at each pole z = − z i , and the residues simplify using the third property to give the final one. f ( z )] ± denotes the expansion of the function f ( z ) in powers of z ∓ .We consider the following correlators that are symmetric functions of the N variables x i , c ( N ) λ ( x ) = N Y i =1 x K N i C h∅| ... ˆ φ ( − x − ) · · · ˆ φ ( − x − N )... | λ i C , (4.34)where K N is an integer to be determined soon. The r.h.s. vanishes unless N + | λ | is even, andwe assume that both N and | λ | are even for simplicity. As seen before, the quantities c ( N ) λ ( x ) aresymmetric functions of the variables x i . Using the first property, we observe that they have a poleof order x K N − N +21 at x = 0. Since there are no other singularities than the points at zero andinfinity, and thanks to the Σ N -invariance, we conclude that the functions c ( N ) λ ( x ) are symmetricpolynomials if K N ≥ N − 2. We choose the minimal value K N = N − 2. Then, using the gradingoperator d C and the fact that α d C ˆ φ ( z ) α − d C = α ˆ φ ( α z ), it is easy to show that the functions c ( N ) λ ( x )are homogeneous of degree | λ | / N ( N − / c ( N ) ∅ ( x ) = N Y i,j =1 i Since W D is a subalgebra of W , it can also berepresented on the Dirac fermion Fock space. From equ. 5.3, we find ρ ( D ) ( W Dm,n ) = I dz iπ z m (cid:0) : ¯ ψ ( z ) ψ ( q n z ) : + : ψ ( z ) ¯ ψ ( q n z ) : (cid:1) . (5.8)This representation of W D has the level ρ ( D ) ( C ) = 1. It can be written as a sum of two Majoranarepresentations with level ρ ( M ) ( C ) = 1 / ψ ( z ) = 1 √ (cid:0) φ (1) ( z ) + iφ (2) ( z ) (cid:1) , ¯ ψ ( z ) = 1 √ (cid:0) φ (1) ( z ) − iφ (2) ( z ) (cid:1) . (5.9) Note also { φ ( z ) , φ ( w ) } = z − δ ( z/w ) , D h∅| φ ( z ) φ ( w ) |∅i D = 1 z − w . (5.6) 28s a result, we find for the generators ρ ( D ) ( W Dm,n ) = I dz iπ z m (cid:0) : φ (1) ( z ) φ (1) ( q n z ) : + : φ (2) ( z ) φ (2) ( q n z ) : (cid:1) , (5.10)which implies that the Dirac representation decomposes as ρ ( D ) = ρ ( M ) ⊗ ⊗ ρ ( M ) . Horizontal presentation In order to build a Heisenberg algebra, we need to consider a new setof operators acting on the Majorana Fock space, namely¯ W Dm,n = I dz iπ z m : φ ( z ) φ ( − q n z ) : = X r ∈ Z +1 / ( − r +1 / q − ( r +1 / n : φ m − r φ r : . (5.11)These operators form a module for the adjoint action of W D , they satisfy[ W Dm,n , ¯ W Dm ′ ,n ′ ] = ( q m ′ n − ( − m q mn ′ ) (cid:18) ¯ W Dm + m ′ ,n + n ′ + δ m + m ′ q n + n ′ (cid:19) + ( − m ′ q − ( m ′ +1) n ′ ( q m ′ n − ( − m q − mn ′ ) (cid:18) ¯ W Dm + m ′ ,n − n ′ + δ m + m ′ q n − n ′ (cid:19) , [ ¯ W Dm,n , ¯ W Dm ′ ,n ′ ] = (( − m ′ q m ′ n − ( − m q mn ′ ) (cid:18) W Dm + m ′ ,n + n ′ + δ m + m ′ − q n + n ′ (cid:19) + ( − m ′ q − ( m ′ +1) n ′ (( − m ′ q m ′ n − ( − m q − mn ′ ) (cid:18) W Dm + m ′ ,n − n ′ + δ m + m ′ − q n − n ′ (cid:19) . (5.12)In particular, the modes a k = ¯ W k, with k even form the Heisenberg subalgebra [ a k , a l ] = 2 kδ k + l .The operators act on the bosonic field as follows,[ W Dm,n , φ ( z )] = z m (cid:0) q − ( m +1) n φ ( q − n z ) − φ ( q n z ) (cid:1) , [ ¯ W Dm,n , φ ( z )] = z m (cid:0) ( − m +1 q − ( m +1) n φ ( − q − n z ) − φ ( − q n z ) (cid:1) , (5.13)and so [ a k , φ ( z )] = − z k φ ( − z ). Due to the sign flip of the field’s argument, it is necessary todecompose into odd and even modes, φ ( z ) = φ e ( z ) + zφ o ( z ) , φ e ( z ) = X k ∈ Z z − k φ k − / , φ o ( z ) = X k ∈ Z z − k − φ k +1 / . (5.14)Both field φ e ( z ) and φ o ( z ) are even functions of z . They transform as follows under the adjointaction of the Heisenberg modes,[ a k , φ e ( z )] = − z k φ e ( z ) , [ a k , φ o ( z )] = 2 z k φ o ( z ) . (5.15)In addition, they satisfy the anticommutation relations { φ e ( z ) , φ e ( w ) } = { φ o ( z ) , φ o ( w ) } = 0 , { φ e ( z ) , φ o ( w ) } = z − δ ( z /w ) . (5.16)29hese relation are reproduced by the bosonization φ e ( z ) = e Q V − ( z ) V + ( z ) z a , φ o ( z ) = e − Q V − ( z ) − V + ( z ) − z − a , V ± ( z ) = e ± P k> δ k, even z ∓ kk a ± k , (5.17)where the zero modes obey [ a , Q ] = 1. Defining the currents, w Dn ( z ) = (1 − q n ) X m ∈ Z z − m W Dm,n + 2 C, ¯ w Dn ( z ) = (1 + q n ) X m ∈ Z z − m ¯ W Dm,n + 1 , (5.18)they are represented on the fermionic Fock space as (1 − σq n ) zφ ( z ) φ ( σq n z ) with σ = 1 for w n ( z )and − w n ( z ). This formula is bosonized into (1 − σq n )(1 − q n ) X ± q n (1 ∓ z ∓ e ± Q ... e ± P k =0 δ k, even z − kk (1+ q − nk ) a k ... z ± a q ± na + q n (1 ∓ / (1 + σq n ) X ± ... e ± P k =0 δ k, even z − kk (1 − q − nk ) a k ... . (5.19)thus providing the horizontal presentation. The vertical presentation for ˆ D ∞ is similar to the one of ˆ B ∞ as it is possible to define a map φ ± r → ( ± ) r +1 / ˜ φ ± r ± / between the modes (although the zero-mode ˜ φ must be considered separately).The states of the PBW basis can be parameterized either by strict partitions, or by symmetricpartitions, using the bijection sending the columns height µ i of the former to the hook lengths h i = 2 µ i − λ with hook lengths h i ∈ Z + 1, i = 1 · · · ℓ ( λ ) the state | λ i D = φ − h / · · · φ − h d ( λ ) / |∅i D . (5.20)Dual states D h µ | are defined using the adjoint action of the modes φ r = φ †− r on the dual vacuum D h∅| . It produces orthonormal states D h λ || µ i D = δ λ,µ .The action of the operators W Dm,n on the states | λ i D is obtained by a direct computation, usingthe commutation relations 5.13. Fortunately, to express this action we can borrow most of the The similarity between these states and the Schur states in the Dirac representation can be understood byrecalling that in the case of a Majorana fermion, particles and holes are identified and one may see them both asholes. The Schur states are defined as a product of operators ¯ ψ − a i − / ψ − ℓ i − / where the index i runs over theboxes on the diagonal of λ , and a i , ℓ i are the arm and leg length resp. of the box of coordinate ( i, i ). Formally, for aMajorana fermion, ¯ ψ − a i − / ≡ ψ − ℓ i − / ≡ φ − h i / , and the equality of arm and leg lengths imposes a restriction tosymmetric partitions. B ∞ . The result takes an unexpectedly compact form,for m strictly positive W D ,n | λ i D = (1 − q n ) q − n X ∈ λ χ n ! | λ i ,W Dm,n | λ i D = X ρ ∈ R Bm ( λ ) ∪ R HPm ( λ ) (cid:0) q − mn χ − nρ − q − n χ nρ ) (cid:1) ( − r ( ρ ) | λ − ρ i D ,W D − m,n | λ i D = X ρ ∈ A Bm ( λ ) ∪ A HPm ( λ ) (cid:0) χ − nρ − q ( m − n χ nρ (cid:1) ( − r ( ρ ) | λ + ρ i D . (5.21)Generators W Dm,n with m > W Dm, | λ i D = 0 for any m .To obtain the action of the operators ¯ W Dm,n , it is sufficient to replace q n by − q n in the action of W Dm,n . It leads to introduce the extra signs ( − s ( ) = ( − i − j associated to a box ∈ λ . The sign( − s ( ρ ) of a strips ρ ∈ A Bm ( λ ) ∪ R Bm ( λ ) is again the sign of the box in the top left corner, and thesign of a hook pair ρ is ( − s ( ρ ) = ( − k ( ρ ) − m +1 . Then, we find¯ W D ,n | λ i D = − (1 + q n ) q − n X ∈ λ ( − s ( ) χ n ! | λ i , ¯ W Dm,n | λ i D = X ρ ∈ R Bm ( λ ) ∪ R HPm ( λ ) (cid:0) ( − m q − mn χ − nρ + q − n χ nρ ) (cid:1) ( − r ( ρ )+ s ( ρ ) | λ − ρ i D , ¯ W D − m,n | λ i D = X ρ ∈ A Bm ( λ ) ∪ A HPm ( λ ) (cid:0) χ − nρ + ( − m q ( m − n χ nρ (cid:1) ( − r ( ρ )+ s ( ρ ) | λ + ρ i D . (5.22)In particular, for k even positive, a k | λ i D = 2 X ρ ∈ R Bk ( λ ) ∪ R HPk ( λ ) ( − r ( ρ )+ s ( ρ ) | λ − ρ i D ,a − k | λ i D = 2 X ρ ∈ A Bk ( λ ) ∪ A HPk ( λ ) ( − r ( ρ )+ s ( ρ ) | λ + ρ i D . (5.23)As in the other cases, the algebra W D can be supplemented by a grading operator d D such that[ d D , W Dm,n ] = − mW Dm,n . It is represented as d D = X r ∈ Z +1 / r : φ − r φ r : , d D | λ i D = | λ | | λ i D , α d D W Dm,n α − d D = α − m . (5.24)It is also possible to show that [ d D , ¯ W Dm,n ] = − m ¯ W Dm,n , or α d D ¯ W Dm,n α − d D = α − m . The number ofhooks varies by 0 , ± W Dm,n (and the operators ¯ W Dm,n ). Thus, therepresentation is not irreducible but contains two sectors corresponding to partitions with either anodd or an even number of hooks. 31 .3 Symmetric polynomials The discussion in this subsection is parallel to the case of ˆ C ∞ , with the possibility of constructingtwo types of polynomials depending on whether we generalize the bosonic expression 3.21 or theexpression 3.24 as a correlator of Majorana fields. Thus, the symmetric polynomials D λ ( x ), andthe skew-polynomials D λ/µ ( x ) are defined as D λ ( x ) = D h∅| e P k> δ k, even pk ( x ) k a k | λ i D , D λ/µ ( x ) = D h µ | e P k> δ k, even pk ( x ) k a k | λ i D . (5.25)Since the action of a k with k > | λ i D produces a finite sum of states | µ i D with µ ≺ λ ,these polynomials decomposes into a finite sum of products of elementary power sums. Insertingthe grading operator d D , it is easily shown that they are homogeneous polynomials of degree | λ | / | λ | − | µ | ) / k even p k ( x ) D λ ( x ) = X ρ ∈ A Bk ( λ ) ∪ A HPk ( λ ) ( − r ( ρ )+ s ( ρ ) D λ + ρ ( x ) ,k ∂∂p k ( x ) D λ ( x ) = 2 X ρ ∈ R Bk ( λ ) ∪ R HPk ( λ ) ( − r ( ρ )+ s ( ρ ) D λ − ρ ( x ) . (5.26)A priori, these polynomials do not satisfy any q-difference equation that could be related to theaction of either operator W D ,n or ¯ W D ,n .The definition of the polynomials c ( N ) λ ( x ) with N variables also extends to the case of ˆ D ∞ . Forthis purpose, we introduce the notation... φ ( z ) · · · φ ( z N )... = N Y i,j =1 i 2. Unfortunately, the definition of an inductive limit suffers from the same problem since as x N +2 , x N +1 → d ( N +2) λ ( x ) (cid:12)(cid:12)(cid:12) x N +2 = x N +1 =0 = N Y i =1 x i d ( N ) λ ( x ) . (5.29)The simplest polynomial can be evaluated as a Pfaffian, d ( N ) ∅ ( x ) = N Y i,j =1 i The main results presented in this paper are the vertical presentations 3.18, 4.25 and 5.21 for thesubalgebras W X of quantum W ∞ with X = B, C, D . These formulas express the action of thegenerators on the Fock state in a combinatorial manner. They were further used to derive a setof rules and a q-difference equation for a corresponding set of symmetric polynomials. In the caseof W B , these polynomials are Q-Schur functions indexed by strict partitions. Since they can beobtained as a specialization of Hall-Littlewood polynomials at t = − 1, it is natural to wonderwhether Hall-Littlewood polynomials would satisfy a t -deformed q-difference equation. Of course,this equation has to be different from the q = 0 limit of the Macdonald equation they already satisfy.Furthermore, the expressions 3.9, 4.9 and 5.5 of the generators W Xm,n in terms of fermionic/bosonicmodes appears very generic. It is tempting to use the same expression with Jing’s t -fermions [50] todefine a quantum W-algebra. It would be interesting to investigate further the possible connectionsbetween this algebra, the Hall-Littlewood polynomials and the quantum affine algebra of sl (2)obtained in the q → gl (1) algebra.In the case of W C and W D , we were led to introduce two sets of symmetric polynomials for eachalgebra. The first set, denoted C λ ( x ) and D λ ( x ) is built using the action of an Heisenberg algebra onthe Fock states of the representation. The polynomials can be define for infinitely many variables,they obey Pieri-like rules under the multiplication by power-sums. However, they do not satisfyany q-difference equation (a priori). The second sets of polynomials, denoted c ( N ) λ ( x ), d ( N ) λ ( x ), aredefined with finitely many variables N and do obey the q-difference equations 4.38 and 5.32 thatnaturally extend 3.26 obtained for W B . In the case of W C , the polynomials C λ ( x ) and c ( N ) λ ( x )can be related using a formalism based on supersymmetric variables introduced in [10]. We expectthat a similar formalism exists in the case of W D as well. Then, it may be possible to introduce asupersymmetric version of the algebra W X that would act on the correlators of the superfields.Our main motivation for the study of these subalgebras is the application to topological stringtheory. Using both vertical and horizontal presentations, one may be able to define the equivalentof the Awata-Feigin-Shiraishi (AFS) intertwiner that provides the operator form of the refinedtopological vertex [18]. In particular, our hope is that the W B algebra would provide the quantumalgebraic structure missing in the earlier attempts by Foda and Wheeler to introduce a B-typetopological vertex. The latter were based upon the Okounkov-Reshetikhin-Vafa melting crystalpicture [39] and used B-type plane partitions [57] and Jing’s t -fermions [58].Going in a different direction, one could try instead to extend the relation between topologicalstrings and integrable hierarchies [38]. In [40, 41], Nakatsu and Takasaki have shown that a certaintime-deformation of a topological string amplitude is a tau function of the KP hierarchy. It would In fact, this property follows from the intertwining relation obeyed by the vacuum component of the AFSintertwiner (see [42]). Thus, it would seem that we are uncovering different aspects of a bigger picture. 34e instructive to extend their calculation to the BKP, CKP and DKP hierarchies. For this purpose,understanding the connections with symmetric polynomials appears to be an essential ingredient.Finally, the toroidal Yangian of gl (1), which is obtained from the quantum toroidal gl (1) algebrain the degenerate limit q , q → 1, provides the COhomological Hall Algebra of the ADHM quiver[59, 60]. It would be interesting to relate the q → W X subalgebras to the cohomologyof a quiver variety. Acknowledgments The author’s interest in the algebraic structures presented in this article arose from discussions withOmar Foda at the University of Melbourne. Omar had an impressive intuition on this topic, hisunique way of thinking and his invaluable advice are sorely missed, as much as his warm friendship.The author would like to thank also Sasha Garbali, Yutaka Matsuo, Vincent Pasquier and KilarZhang 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. A Bosonic correlators in Schur basis It is an established fact that skew-Schur polynomials can be written as correlators in the free bosonFock space, s λ/µ ( x ) = h λ | Γ − ( x ) | µ i = h µ | Γ + ( x ) | λ i , Γ ± ( x ) = exp X k> p k ( x ) k a ± k ! . (A.1)In the particular case µ = ∅ , we recover the formula 2.17. In this appendix, we use the corre-spondence between symmetric polynomials and the free boson Fock space to derive a proof of thisformula from well-known identities obeyed by skew-Schur polynomials. Recall that the Schur poly-nomials s λ ( x ) correspond to the bosonic states | λ i x under this correspondence, where we indicateby the label x the states belonging to the Fock space associated to polynomials in variables x (wewill be considering several Fock spaces).As a warm-up, we consider the following formula from [8], X λ s λ ( x ) s λ ( y ) = Y i,j (1 − x i y j ) − = exp X k> k p k ( x ) p k ( y ) ! . (A.2)Taking the correspondence along the y Fock space, we have X λ s λ ( x ) | λ i y = exp X k> k p k ( x ) a − k ! |∅i y = Γ − ( x ) |∅i y ⇒ s λ ( x ) = h λ | Γ − ( x ) |∅i . (A.3)35ext, we consider the formula [8] (page 71) X λ,µ s λ/µ ( x ) s λ ( y ) s µ ( z ) = Y i,j (1 − x i y j ) − × Y i,j (1 − y i z j ) − = exp X k> k ( p k ( x ) p k ( y ) + p k ( y ) p k ( z )) ! . (A.4)Taking the correspondence for the variables y and z , we get X λ,µ s λ/µ ( x ) | λ i y ⊗ | µ i z = exp X k> k ( p k ( x ) a − k ⊗ a − k ⊗ a − k ) ! |∅i y ⊗ |∅i z . (A.5)Inserting the identity in the y -space, and projecting on y h λ | ⊗ z h µ | , we find s λ/µ ( x ) = X η y h λ | e P k> k p k ( x ) a − k | η i y (cid:16) y h η | ⊗ z h µ | e P k> k a − k ⊗ a − k |∅i y ⊗ |∅i z (cid:17) . (A.6)On the other hand, the identity A.2 also implies e P k> k a − k ⊗ a − k |∅i y ⊗ |∅i z = X λ | λ i y ⊗ | λ i z , (A.7)and, as a result, a delta function δ η,µ appears in the RHS of A.6, leading to s λ/µ ( x ) = h λ | e P k> k p k ( x ) a − k | µ i = h λ | Γ − ( x ) | µ i . (A.8)Moreover, since Γ + ( x ) = Γ − ( x ) † , the second equality also holds.It is worth mentioning that starting instead from the relations X λ s λ ( x ) s λ ′ ( y ) = Y i,j (1 + x i y j ) = exp − X k k p k ( − x ) p k ( y ) ! , X λ,µ s λ/µ ( x ) s λ ′ ( y ) s µ ′ ( z ) = Y i,j (1 + x i y j ) × Y i,j (1 − y i z j ) − = exp X k> k ( − p k ( − x ) p k ( y ) + p k ( y ) p k ( z )) ! . (A.9)we can show similar formulas involving the transposed Young diagrams λ ′ , s λ/µ ( x ) = h λ ′ | Γ − ( − x ) − | µ ′ i = h µ ′ | Γ + ( − x ) − | λ ′ i , (A.10) The first relation is given in [8] (page 65) while the second one can be derived from A.4 using the homomorphism[8] (page 42) ω on the x variables. The latter acts as ω ( s λ/µ ) = s λ ′ /µ ′ on skew-Schur polynomials and ω ( p k ) =( − k − p k on elementary power sums. Derivation of the q-difference equations B.1 Schur case We derive in this appendix the q-difference equation 2.22 obeyed by the Schur polynomials fromthe diagonal action of the modes W ,n in the vertical presentation. Inserting these operators in ther.h.s. of equ. 2.17, and using 2.14 we find h∅| e P k> pk ( x ) k a k W ,n | λ i = (1 − q n ) q − n X ∈ λ χ n ! s λ ( x ) . (B.1)On the other hand, the operators W ,n are the zero modes of the currents w n ( z ) defined in 2.11.Using the bosonized version of the current, we find after normal-ordering that e P k> pk ( x ) k a k ((1 − q n ) W ,n + 1) = I dz iπz Y i − q n x i z − x i z ... e P k> pk ( x ) k a k w n ( z )... . (B.2)The contour of the integral can deformed on the sphere, thus picking up residues at z = x − i andinfinity, and we find(1 − q n ) X i =1 Y j = i − q n x j /x i − x j /x i e P k> x − ki (1 − q nk ) a − kk e P k> ( p k ( x )+ x ki ( q − nk − ) akk − I ∞ dz iπz Y i − q n x i z − x i z ... e P k> pk ( x ) k a k w n ( z )... . (B.3)The negative modes disappear after projection on h∅| , and the contribution at infinity simplifiesdrastically since the operator has no longer any positive power of z . As a result, we get h∅| e P k> pkk a k (cid:18) W ,n + 1 − q nN − q n (cid:19) = N X i =1 Y j = i x i − q n x j x i − x j h∅| e P k> ( p k ( x )+ x ki ( q − nk − ) akk , (B.4)where N is the number of variables which is used as a cut-off. The shift of p k ( x ) in the exponent ofthe i th term corresponds to the replacement of the variable x i by q − n x i . Following [8], we denotethis operation by T q − n ,x i . Projecting on the state | λ i and combining the result with the relationB.1, we establish the q-difference equation 2.22. Power sum p The simplest (non-trivial) Schur polynomial is the one associated to the partition1 containing a single box, it is simply the first symmetric power sum p ( x ). This polynomial shouldsatisfy the q-difference equation 2.22, which reads in this case q n N X i =1 Y j = i x i − q n x j x i − x j T q − n ,x i − N X i =1 q ni ! p ( x ) = (1 − q n ) p ( x ) . (B.5)37t is instructive to check this equation against a direct calculation. Using T q − n ,x i p ( x ) = ( q − n − x i + p ( x ), it can be rewritten in the form (cid:18) S ( x ) + q n − q n S ( x ) p ( x ) − q n − q nN (1 − q n ) (cid:19) = p ( x ) , with S k ( x ) = N X i =1 x ki Y j = i x i − q n x j x i − x j . (B.6)The sums S k ( x ) can be computed using Jacobi identities, but we prefer to present here an alternativederivation. We introduce the function S ( z ) = Y i z − q n x i z − x i = 1 + (1 − q n ) X i x i z − x i Y j = i x i − q n x j x i − x j , (B.7)where the second equality is found by decomposition over the poles. Considering the value S (0)and the subleading term in the large z asymptotic expansion, we can show that S ( x ) = 1 − q nN − q n , S ( x ) = p ( x ) . (B.8)Plugging this back into the l.h.s. of the equation B.6 we find p , which shows that the first powersum does indeed obey the q-difference equation B.5. B.2 B-case The derivation of the q-difference equation for b λ ( x ) is very similar to the Schur case and uses thebosonization of the Majorana fermion. We first introduce the equivalent of the currents w n ( z ) forthe subalgebra W B , w Bn ( z ) = (1 − q n ) X m ∈ Z z − m W Bm,n + 2 C. (B.9)In the Majorana representation, they read w Bn ( z ) = (1 − q n ) ˜ φ ( z ) ˜ φ ( − q n z ) which is bosonized into w Bn ( z ) = 12 (1 + q n )... e − P k ∈ Z δ k, odd z − kk (1 − q − nk ) a k ... . (B.10)We also introduce the shortcut notationΓ B ( x ) = e P k> δ k, odd pk ( x ) k W Bk, ⇒ b λ ( x ) = B h∅| Γ B ( x ) | λ i B , b ∗ λ ( x ) = 2 B h | Γ B ( x ) | λ i B . (B.11)After these preliminaries, we consider the insertion of the operator W B ,n in the correlator 3.21and use its diagonal action 3.18 to show that B h∅| Γ B ( x ) W B ,n | λ i B = (1 − q n ) X ∈ λ − ( q − n χ n + χ − n ) − b λ ( x ) . (B.12)38o derive the q-difference equation, we use the fact that the operators W B ,n correspond to thezero-modes of the current w Bn ( z ). From the bosonic expressions B.10 and B.11, we compute thenormal-ordering I dz iπz B h∅| Γ B ( x ) w Bn ( z ) | λ i B = 12 (1 + q n ) I dz iπz Y i (1 + x i z )(1 − q n x i z )(1 − x i z )(1 + q n x i z ) B h∅| ... e P k> δk, odd k ( p k ( x ) − z − k + q − nk z − k ) a k ... | λ i B . (B.13)Negative modes were removed from the correlator since they annihilate the dual vacuum h∅| . De-forming the contour on the sphere, we pick up residues at z = x − i , − q − n x − i and infinity,12 (1 + q n ) B h∅| ... e P k> δk, odd k p k ( x ) a k ... | λ i B + (1 − q n ) X i B ( n ) i ( x ) B h∅| ... e P k> δk, odd k ( p k ( x ) − x ki + q − nk x ki ) a k ... | λ i B − (1 − q n ) X i B ( − n ) i ( x ) B h∅| ... e P k> δk, odd k ( p k ( x )+ q nk x ki − x ki ) a k ... | λ i B (B.14)with the residues B ( n ) i ( x ) given in 3.27. This expression can be rewritten as the action of theq-difference operators on the symmetric polynomials I dz iπz B h∅| Γ B ( x ) w Bn ( z ) | λ i B = (1 − q n ) "X i (cid:16) B ( n ) i ( x ) T q − n ,x i − B ( − n ) i ( x ) T q n ,x i (cid:17) + 12 1 + q n − q n b λ ( x ) . (B.15)Combining this relation with B.12, we find the q-difference equation 3.26. The derivation extendsto b ∗ λ ( x ) since the property B h | a k = 0 for k < Power sum p For b = p , the q-difference equation 3.26 reads X i (cid:16) B ( n ) i ( x ) T q − n ,x i − B ( − n ) i ( x ) T q n ,x i (cid:17) p ( x ) = q − n (1 − q n ) p ( x ) . (B.16)We can check this formula against a direct computation. We have X i (cid:16) B ( n ) i ( x ) T q − n ,x i − B ( − n ) i ( x ) T q n ,x i (cid:17)! p ( x ) = − (1 − q − n ) S B ( x ) + S B ( x ) p ( x ) , with S B ( x ) = X i (cid:16) B ( n ) i ( x ) − B ( − n ) i ( x ) (cid:17) , S B ( x ) = X i (cid:16) x i B ( n ) i ( x ) + q n x i B ( − n ) i ( x ) (cid:17) . (B.17)In order to compute the sums S Bk ( x ). we consider the function B ( z ) = Y i ( z + x i )( z − q n x i )( z − x i )( z + q n x i ) = 1 + 2 1 − q n q n X i (cid:18) x i z − x i B ( n ) i ( x ) + q n x i z + q n x i B ( − n ) i ( x ) (cid:19) , (B.18)at z = 0 and z = ∞ . We find that S B ( x ) = 0 and S B ( x ) = (1 + q n ) p ( x ). Thus, we recover indeedthe q-difference equation B.16 from equ. B.17. 39 .3 C-case Once again, we exploit the fact that the operators W C ,n are diagonal on the states | λ i C and coincidewith the zero modes of the currents w Cn ( z ). From the expression w Cn ( z ) = − (1 − q n ) z ˆ φ ( z ) ˆ φ ( − q n z ),we compute the commutator[ w Cn ( z ) , ... ˆ φ ( − x − ) · · · ˆ φ ( − x − N )...]= (1 − q n ) q − n N X i =1 δ ( − q n x i z ) Y j = i ( − x − i − x − j ) ˆ φ ( z )... ˆ φ ( − x − ) · · · ✘✘✘✘✘ ˆ φ ( − x − i ) · · · ˆ φ ( − x − N )... − (1 − q n ) N X i =1 δ ( x i z ) Y j = i ( − x − i − x − j ) ... ˆ φ ( − x − ) · · · ✘✘✘✘✘ ˆ φ ( − x − i ) · · · ˆ φ ( − x − N )... ˆ φ ( − q n z ) (B.19)Using the first and second properties, we can insert ˆ φ ( z ) and ˆ φ ( − q n z ) inside the symbols ... · · · ....Doing so, we reintroduce a dependence on the coordinate x i inside the correlator, but the argumentof the corresponding field has an extra factor of q ± n . Thus, we end up with[ w Cn ( z ) , ... ˆ φ ( − x − ) · · · ˆ φ ( − x − N )...]= (1 − q n ) N X i =1 δ ( − q n x i z ) q − n Y j = i x i + x j x i + q − n x j T q n ,x i − δ ( x i z ) Y j = i x i + x j x i + q n x j T q − n ,x i ! ... ˆ φ ( − x − ) · · · ˆ φ ( − x − N )...(B.20)Integrating over z , we deduce the commutator of the zero mode[ W C ,n , ... ˆ φ ( − x − ) · · · ˆ φ ( − x − N )...]= N X i =1 q − n Y j = i x i + x j x i + q − n x j T q n ,x i − Y j = i x i + x j x i + q n x j T q − n ,x i ! ... ˆ φ ( − x − ) · · · ˆ φ ( − x − N )... (B.21)Projecting on the states C h∅| and | λ i C , and using the vertical action 4.25, we deduce the q-differenceequation N X i =1 Y j = i x i + x j x i + q n x j T q − n ,x i − q − n Y j = i x i + x j x i + q − n x j T q n ,x i ! C h∅| ... ˆ φ ( − x − ) · · · ˆ φ ( − x − N )... | λ i C = (1 − q n/ ) q − n/ X ∈ λ ( ˆ χ n + q − n/ ˆ χ − n ) ! C h∅| ... ˆ φ ( − x − ) · · · ˆ φ ( − x − N )... | λ i C . 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