Abstract
We construct the generalized \(\beta \) and (q, t)-deformed partition functions through W representations, where the expansions are respectively with respect to the generalized Jack and Macdonald polynomials labeled by N-tuple of Young diagrams. We find that there are the profound interrelations between our deformed partition functions and the 4d and 5d Nekrasov partition functions. Since the corresponding Nekrasov partition functions can be given by vertex operators, the remarkable connection between our \(\beta \) and (q, t)-deformed W-operators and vertex operators is revealed in this paper. In addition, we investigate the higher Hamiltonians for the generalized Jack and Macdonald polynomials.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Nekrasov functions have attracted much attention, which are the generalizations of hypergeometric series originally introduced to regularize the integrals over instanton moduli space [1, 2]. The exact partition functions were obtained as a perturbative sum over instanton sectors of coupled contour integrals. The evaluation of these integrals by the Cauchy theorem produces a sum over residues in one-to-one correspondence with the set of boxes of Young diagrams. Alday, Gaiotto and Tachikawa conjectured the relations (AGT conjectures or relations) between 4d Nekrasov partition functions with SU(2) gauge groups and 2d Liouville theory [3]. It can be extended to 5d and 6d SYM theories [4, 5] and SU(N) quiver gauge theories [6]. The 4d AGT relations with SU(N) gauged group can be described by a special orthogonal basis (the AFLT basis) of the highest weight states of the \(W_N\otimes Heisenberg\) algebra [7,8,9,10] which is labeled by N-tuple of Young diagrams. With the help of AFLT bases, the Nekrasov partition functions for U(N) gauge theory with 2N fundamental matters were given by the two-point correlation function of a particular vertex operator [8,9,10,11,12,13]. The AFLT basis, sometimes called the fixed-point basis in the geometric representation [14, 15], can be constructed by the algebraic approaches. The generalized Jack polynomials (GJP) are defined by the centrally-extended spherical degenerate double affine Hecke algebra (\(\mathbf {SH^c}\)) [9, 10, 16], and the generalized Macdonald polynomials (GMP) are defined by the quantum toroidal \(\mathfrak {gl}_1\) algebra [12, 15, 17]. The GJP and GMP can be regarded as the 4d and 5d AFLT bases, respectively. The proofs of AGT relations which are based on the Dotsenko–Fateev (DF) integral representation of the conformal blocks [18] have been discussed [7, 8, 19,20,21,22,23,24]. By decomposing the DF integral in terms of GJP or GMP, the proof of AGT relations transforms into computing the corresponding Selberg integral averages. Moreover, this decomposition is naturally identified with the corresponding topological string amplitude, computed using the topological vertex technique, especially in the 5d case [13, 22, 23].
The Nekrasov partition functions can be related to matrix models [25,26,27,28,29,30,31]. They emerge in matrix models as a straightforward implication of superintegrability, factorization of peculiar matrix model averages [31]. W-representations of matrix models realize the partition functions by acting on elementary functions with exponents of the given W-operators [32,33,34,35,36]. They are conducive to analyze the structures of matrix models. The Hurwitz–Kontsevich (HK) matrix model can be used to describe the Hurwitz numbers and Hodge integrals over the moduli space of complex curves [37, 38]. There is the W-representation for this superintegrable matrix model, where the W-operator is the Hurwitz operator. Recently, the partition function hierarchies were presented by the expansions with respect to the Schur functions via the W-representations, where the W-operators are given by the nested commutators in which the Hurwitz operator plays a fundamental role [39]. For the negative branch of hierarchy, it gives the \(\tau \)-functions of the KP hierarchy. It can be described by the two-matrix model that depends on two (infinite) sets of variables and an external matrix [40,41,42]. For the \(\beta \)-deformed partition function hierarchies in [39], their integral realizations and ward identities were presented by means of \(\beta \)-deformed Harish–Chandra–Itzykson–Zuber integral [43, 44]. Furthermore, the W-operators in the positive branch of hierarchies can be related to the many-body systems [45].
Much interest has been attributed to (q, t)-deformed matrix models [35, 46,47,48,49,50,51,52,53,54,55]. For some (q, t)-deformed matrix models, the (q, t)-deformation was carried out by substituting the Schur functions in superintegrability expansions by the Macdonald polynomials [35, 50, 51]. Ding–Iohara–Miki algebra (DIM) or quantum toroidal \(\mathfrak {gl}_1\) algebra is a double centrally expanded and double quantum parameters deformed algebra [56,57,58] and has rich representation theories [15, 17, 59,60,61]. Based on DIM, some (q, t)-deformed partition functions have been constructed by different approaches [48, 49, 54, 55]. Network matrix models are naturally built from the Seiberg-Witten integrable system. Mironov et al. worked out the connection between a large class of network matrix models associated with toric diagrams and the DIM algebra [48, 49]. Recently, the (q, t)-deformed partition functions were presented through W-representations, where the W-operators are determined by the cut-join rotation operator \({\hat{O}}\) [54]. The construction of \({\hat{O}}\) can be done in terms of a Fock representation of DIM algebra or the Macdonald difference operators.
The (centrally-extended) elliptic Hall algebra defined as Hall algebra of elliptic curve [16, 62, 63] may be thought as the stable limit of spherical double affine Hecke algebra (sDAHA) and isomorphic to DIM. These generators are \({\mathbb {Z}}^2\) graded so that it is easier to describe the commutation relations of generators and the \(SL(2,{\mathbb {Z}})\) automorphism (Miki automorphism). Motivated by the recent progress in W-representations, in this paper, we will construct the generalized \(\beta \) and (q, t)-deformed W-operators based on the \(\mathbf {SH^c}\) and elliptic Hall algebra, respectively. Our goal is to present the generalized \(\beta \) and (q, t)-deformed partition functions of the expansions of the GJP and GMP via W-representations and to establish the connection between them and the Nekrasov partition functions.
This paper is organized as follows. In Sect. 2, we first construct the \(\beta \)-deformed W-operators which are closely related to the Calogero–Sutherland (CS) models based on the Fock representation (symmetric functions representation) of the algebra \(\mathbf {SH^c}\). Then we construct the generalized \(\beta \)-deformed W-operators from the N-Fock representation of the algebra \(\mathbf {SH^c}\) and present the partition functions of the expansions with respect to GJP through W-representations. We further point out the interrelations between our partition functions and the 4d Nekrasov partition functions. In Sect. 3, we extend the W-operators and partition functions to the generalized (q, t)-deformed versions with the help of the elliptic Hall algebra. We present the higher Hamiltonians for GMP. Moreover we consider the semi-classical limits and give the higher Hamiltonians for GJP. Finally, it is shown that there are the interrelations between the generalized (q, t)-deformed partition functions and the 5d Nekrasov partition functions. We explore the deep connection between our deformed W-operators and vertex operators. We end this paper with the conclusion in Sect. 4.
2 Generalized \(\beta \)-deformed partition functions and 4d Nekrasov partition functions
2.1 Algebra \(\mathbf {SH^c}\), \(\beta \)-deformed Hurwitz operator and deformed CS operators
Let us recall the algebra \(\mathbf {SH^c}\) and its polynomial representation [9, 16]. The algebra \(\mathbf {SH^c}\) is defined by generators \(\{D_{r,s},{\textbf{c}}_s\}_{r\in {\mathbb {Z}},s\in {\mathbb {Z}}_{\ge 0}}\) with commutation relations
where \(D_{0,0}=0\), \({\textbf{E}}_0={\textbf{c}}_0\) and \({\textbf{E}}_k\) is a nonlinear combination of \(\{D_{0,l},{\textbf{c}}_l\}\) as follows
where
For convenience, we take the center charges \({\textbf{c}}_l=\delta _{0,l}\) in this paper. Thus the algebra \(\mathbf {SH^c}\) is generated by the triples \((D_{0,2},D_{1,0},D_{-1,0})\) here.
The algebra \(\mathbf {SH^c}\) is isomorphic to the affine Yangian of \(\mathfrak {gl}_1\) (\({\mathcal {Y}}\{\widehat{\mathfrak {gl}_1}\}\)) in [64] via the following assignments
and the specialization of \(\mathbf {SH^c}\) at \(\beta =1\) is isomorphic to the universal enveloping algebra of the Witt algebra \(w_{1+\infty }\).
Similar to the \({\mathcal {Y}}\{\widehat{\mathfrak {gl}_1}\}\), \(\mathbf {SH^c}\) admits the following spectral shift automorphism parametrized by \(a\in {\mathbb {C}}\)
Therefore the free boson representations of these generators \(\{D_{0,1},D_{0,2},D_{\pm l,0}\}\) for \(l\ge 1\) can be defined by power sum variables
The actions of the operators \(\rho _a(D_{0,l})\) on Jack functions are
The operator \(\rho _0(D_{0,2})\) is closely related to the trigonometric Calogero–Sutherland (tCS) operator as follows:
There are also the geometric interpretations for \(\rho _0(D_{0,2})\) [15, 65, 66]. A non-linear relation between higher Hamiltionians of tCS system and operators \(\rho _a(D_{0,k})\) with \(k\ge 0\) was discussed in Ref. [16].
In terms of the operator \(\rho _0(D_{0,2})\), the \(\beta \)-deformed Hurwitz operator [39] can be expressed as
where L is the order of integrated Hermitian matrices.
With the help of the commutation relations (1), we construct the operators \(W_{\pm n}(a;p)\) by the nested commutators
![](http://media.springernature.com/lw379/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-024-13040-w/MediaObjects/10052_2024_13040_Equ9_HTML.png)
where the operators \(W_n(a;p)\) in the positive branch are conjugate to \(W_{-n}(a;p)\) for the scaling product \(\langle \cdot ,\cdot \rangle _{\beta }\), i.e., \(\langle W_n f,g\rangle _{\beta }=\langle f,W_{-n}g\rangle _{\beta }\) for any symmetric functions f and g. We should remark that \(W_{\pm n}(a;p)\) depend on the spectral parameter a and \([W_{\pm n}(a;p),W_{\pm m}(a;p)]=0\) for \(n,m\ge 1\).
Let us list several operators
The operators \(W_{n}(a;p)\) can be related to the many-body systems.
(i) Let us take \(a=L\beta \), \(p_k=\sum _{i=1}^{L}x_i^k\) and denote \(\hat{H}_n=\beta ^{n-1}W_n(a;p)\) for \(n\le L\), then we obtain the rational Calogero–Sutherland (rCS) operators derived in Ref. [45]
Taking the similarity transformation
with \(\Delta (x)=\prod _{i<j}^{L}(x_i-x_j)\), it gives the higher Hamiltonians of the rCS model with L particles.
(ii) Let us take \(a=L\beta -M\), \(\bar{p}_k=\sum _{i=1}^{L}x_i^k-\beta ^{-1}\sum _{j=1}^{M}y_j^k\) and denote \(\bar{H}_n=\beta ^{n-1}W_{n}(a;\bar{p})\) for \(n\le \text {min}\{L,M\}\), then we obtain the deformed rCS operators
The higher Hamiltonians of U(L|M)-deformed rCS model [67] are given by the similar transformation
with
\(\bar{{\mathscr {H}}}_1\) and \(\bar{{\mathscr {H}}}_2\) are the momentum and Hamiltonian, respectively,
By means of the operators (9), we give the partition functions through W-representations
where \(\textsf{Jack}_{\lambda }\) is the integral form of Jack polynomial [68].
When setting \(a=N\beta \) and \(p_k=\sum _{i=1}^{N}x_i^k\), \(Z_{-n}(a;p)\) and \(Z_{n}(a;p|g)\) recover the negative and positive branch of \(\beta \)-deformed partition functions in Ref. [39]. Recently, their integral realizations and ward identities were presented [43, 44].
Let us consider the case of \(a=L\beta -M\) and \(p_k=\sum _{i=1}^{L}x_i^k-\beta ^{-1}\sum _{j=1}^{M}y_j^k\) in \(Z_{\pm n}(a;p)\). There is the super version of the \(\beta \)-deformed Cauchy identity
where \(\textsf{SJ}_\lambda \{x;y\}:=\textsf{Jack}_{\lambda }\{p\}\) is the super Jack polynomial and \(\mathcal{S}\mathcal{P}\) is the set of the fat (L, M)-hook Young diagrams [67]. By replacing \(\textsf{Jack}_{\lambda }\{p\}\) in (17a) and (17b) with \(\textsf{SJ}_\lambda \{x;y\}\) and restricting the sum range of Young diagrams \(\lambda \) and \(\mu \) to \(\mathcal{S}\mathcal{P}\), it gives the \(\beta \)-deformed partition functions [69].
The \(\beta \)-deformed ABJ-like model is given by [70]
It can be interpreted as the matrix model corresponding to an integral over supermatrices in the algebra of the supergroup U(L|M).
It is known that the Virasoro constraints of (19) are identical to that ones of the \(\beta \)-deformed Gaussian–Hermitian model just by replacing the latter’s rank L with the effective rank \(L_{eff}=L-\beta ^{-1}M\). Thus there is a constraint equation
From (20), we can confirm that the \(\beta \)-deformed ABJ-like model (19) is the integral realization of \(Z_{-2}(L_{eff};\bar{p})\) in (17a)
where \(c_{\emptyset }=\tau _{L,M}|_{\bar{p}=0}\).
2.2 Generalized \(\beta \)-deformed partition functions
The algebra \(\mathbf {SH^c}\) is equipped with the topological coproduct [16]
and
Note that there could exist the coproduct structure for \(D_{0,l}\), \(l\ge 3\), however it is too complicated to give an explicit formula.
Let \(\vec {a}=(a_1,\ldots ,a_N)\) and \(\vec {p}=(p^{(1)},\ldots ,p^{(N)})\) be the N-tuples of complex parameters and time variables, respectively, and \(\rho _{\vec {a}}^{N}=\prod _{i=1}^{N}\otimes \rho _{a_i}\circ \mathbf {\Delta }^{(N-1)}\). Then there are the N-Fock representation for \(\mathbf {SH^c}\)
where \(p_n^{(k)}={\textbf{1}}\otimes \cdots \otimes {\textbf{1}}\otimes \overset{k -th }{\overbrace{p_n}} \otimes {\textbf{1}}\otimes \cdots \otimes {\textbf{1}}\) with \(k=1,\ldots ,N.\)
Let us denote \({\mathbb {W}}(\vec {a})=\rho _{\vec {a}}^{N}(D_{0,2})\) and \(\vec {\lambda }=(\lambda ^1,\ldots ,\lambda ^N)\) as an N-tuple of Young diagrams. The GJP \(J_{\vec {\lambda }}\{\vec {a};\vec {p}\}\) are defined as
where \(v^{\vec {\mu }}_{\vec {\lambda }}(\vec {a})\) is a complex matrix, “\({<^R}\)” is the partial ordering on the set of N-tuple of Young diagrams [12] and
especially, we have
Since the non-symmetric term \(\sum _{k<l}np_n^{(l)}\frac{\partial }{\partial p_n^{(k)}}\) in \(W(\vec {a};\vec {p})\) is conjugate to \(\sum _{k>l}np_n^{(l)}\frac{\partial }{\partial p_n^{(k)}}\), it is natural to define the dual polynomials \(J^*_{\vec {\lambda }}\) by reversing the order of time variables
Then we have
It implies the Cauchy completeness identity for GJP
Let us introduce the operators \({\mathbb {E}}_n\) and \({\mathbb {F}}_n\) for \(n\ge 1\) as follows:
where
\(\textrm{ad}_f\ g:=[f,g]\), \(\vec {m}=(m_1,m_2,\cdots )\) and \(\vec {a}^{r}=(a_1+m_r,\ldots ,a_N+m_r)\).
In terms of \({\mathbb {E}}_n\) and \({\mathbb {F}}_n\), we construct the generalized \(\beta \)-deformed W-operators with N-tuple of time variables as follows:
where
It is clear that \({\mathbb {E}}_n=W_{-1,n}(\vec {a};\vec {m})\), \({\mathbb {F}}_n=W_{1,n}(\vec {a};\vec {m})\), \({\mathbb {W}}(\vec {a}^r)=W_{0,2}(\vec {a}^r)\) for \(r\ge 1\) and \(W_{l,nl}(\vec {a};\vec {m})\) are conjugate to \(W_{-l,nl}(\vec {a};\vec {m})\).
When restricted to the \(N=1\) case, the operators \(W_{\pm l,nl}(\vec {a};\vec {m})\) degenerate to \(H^{(n)}_{\pm l}\) in Ref. [41] which are labeled by two indices, l is called the grading and \(nl+1\) is the spin. A noteworthy property is that the \(H^{(n)}_{\pm l}\) can be embedded in the \(W_{1+\infty }\) algebra.
We observe that the explicit computation of actions of the operators \(W_{\pm l,nl}(\vec {a};\vec {m})\) (33) on GJP become rather cumbersome. To overcome this problem, we make use of the technique of cut-and-join rotation operators [38, 42, 45, 54]. Let us introduce the generalized cut-and-join rotation operator
such that
With the help of the recursive formulas
and the similar relations for \(W_{-l,nl}(\vec {a};\vec {m};\vec {p})\), the W-operators (33) can be expressed as
The actions of the operators (37) and \(W_{0,n}(\vec {a})\) in (33c) on GJP are
where the coefficients \(C_{\vec {\lambda }}^{\vec {\mu }}\) and \(\bar{C}_{\vec {\lambda }}^{\vec {\mu }}\) are given by
Due to the desired W-operators, we construct the generalized \(\beta \)-deformed partition functions
where the skew polynomials \(J_{\vec {\lambda }/\vec {\mu }}\) are defined by
for any symmetric function f.
2.3 4d Nekrasov partition functions
The 4d Nekrasov partition function for the U(N) theory with \(N_f=2n\) fundamental hypermultiplets is given by [1,2,3]
where
The partition function (42) depends on the mass vectors \(\vec {m}^+\) and \(\vec {m}^-\) for fundamental and anti-fundamental respectively with n components. Note that we take the supergravity background parameters \((\epsilon _1,\epsilon _2)=(-\beta ,1)\) here.
The symmetric function space is isomorphic to the boson Fock space by the means of
such that
where \([\hat{a}_n,\hat{a}_m]=n\beta ^{-1}\delta _{m+n,0}\).
For the vertex operator [71]
it satisfies a Pieri-type formula
For the 4d U(1) Nekrasov partition function with \({\mathcal {N}}_{f}=2\) and \(z=-1\), it can be written in terms of the vertex operator as
More generally, there exists the vertex operator \(\hat{V}(x)={\tilde{V}}_HV_W\) [8, 9], where \(V_W\) is the vertex operator for \(W_N\) algebra and \({\tilde{V}}_H\) describes the contribution of U(1) factor, such that
Similarly, in terms of this vertex operator \(\hat{V}(x)\), one can give the 4d U(N) Nekrasov partition function with \({\mathcal {N}}_{f}=2N\) fundamental matters
where \(m^{+}_i=x_1+b_i\), \(m^{-}_i=x_2-c_i\) for \(i=1,\ldots ,N\).
Let us turn to our partition functions (40a). We observe that by removing the generalized Jack polynomials from (40a) and taking \(m_i=1-\beta -m^{+}_i\), \(m'_i=m^{-}_i\) for \(i=1,2,\ldots ,n\), the remains match with the 4d Nekrasov partition functions (42). An interesting point about this observation is that now we can establish the connection between these two partition functions
Then we make the explicit connection between W-operators (37) and the vertex operator \(\hat{V}(x)\) from (49) and (50)
where \(A_l=(-1)^{Nl}\beta l^{-1}\) and \(m_i=1-\beta -b_i-x_1\), \(m'_i=x_2-c_i\) for \(i=1,\ldots ,N\).
Let us denote \(W_{Dessin}(u,v)\) as \(W_{-1,2}(\vec {a};\vec {m})\) in (33a) with \(\beta =1,(\vec {a};\vec {m})=(0;u,v)\), then
and the corresponding partition function with w-representation and its character expansions are given by
which is nothing but the dessins d’enfant tau-function [72].
It is known that \(Z_{Dessin}=e^{{\mathcal {F}}_{Dessin}(s,u,v)}\), where the free energy is defined as the generating series of weighted count of labeled dessins d’enfants
Comparing (42) with (53) and (54), we denote
then we have
It means that the free energy of the U(1) instanton partition function with \({\mathcal {N}}_f=2\) and \(\beta =1\) is given by the specific part of (connected) Belyi fat graphs.
3 Generalized (q, t)-deformed partition functions and 5d Nekrasov partition functions
3.1 Elliptic Hall algebra and higher Hamiltonians for GMP
Let us recall the elliptic Hall algebra \(\hat{{\mathcal {E}}}\) [62, 63] which is K-algebra generated by elements \(\{{\textbf{u}}_{k,l}\}_{k,l\in {\mathbb {Z}}}\) and centers \(({\textbf{c}}_{0,1},{\textbf{c}}_{1,0})\), modulo the following relations
![](http://media.springernature.com/lw301/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-024-13040-w/MediaObjects/10052_2024_13040_Equ66_HTML.png)
![](http://media.springernature.com/lw360/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-024-13040-w/MediaObjects/10052_2024_13040_Equ67_HTML.png)
![](http://media.springernature.com/lw382/springer-static/image/art%3A10.1140%2Fepjc%2Fs10052-024-13040-w/MediaObjects/10052_2024_13040_Equ68_HTML.png)
where \(\kappa _s=(1-q^s)(1-t^{-s})(1-(t/q)^s)\) and
for all \((k,l)\in {\mathbb {Z}}^2\setminus \{0,0\}\) with \(\text {gcd}(k,l)=1\). The \(\text {gcd}(a,b)=1\) represents that the positive integers a and b are coprime.
\(\hat{{\mathcal {E}}}\) is \({\mathbb {Z}}^2\) graded and has a natural \(SL(2,{\mathbb {Z}})\) automorphism
It can be generated by the fundamental elements \({\textbf{u}}_{0,\pm 1},{\textbf{u}}_{\pm 1,0},{\textbf{c}}_{0,1}\) and \({\textbf{c}}_{1,0}\).
The Cartan generators \(\theta _{0,\pm s}\) and \(\theta _{\pm s,0}\) are generated by
\(\hat{{\mathcal {E}}}\) is isomorphic to the quantum toroidal algebra of \(\mathfrak {gl}_1\) in [59] via the following assignments
We take the centers \(({\textbf{c}}_{0,1},{\textbf{c}}_{1,0})=(1,(t/q)^{1/2})\) in this paper. There is the boson Fock representation of \(\hat{{\mathcal {E}}}\) by power sum variables [63, 73] such as
and the actions
where u is a nonzero complex parameter, \(\textsf{Mac}_\lambda \{p\}\) are integral form of the Macdonald polynomials and \({\textbf{C}}_{\lambda }^{(n)}=\sum _{(i,j)\in \lambda }(q^{j-1}t^{1-i})^n-\frac{1}{(1-q^n)(1-t^{-n})}\) for \(n\ne 0\).
\(\hat{{\mathcal {E}}}\) is equipped with the topological coproduct structure [16]
We denote \(\vec {u}=(u_1,\ldots ,u_n)\) as an N-tuple of nonzero complex parameters and \(\rho _{\vec {u}}:=\prod _{k=1}^{N} \otimes \rho _{u_k}\circ \Delta ^{(N-1)}\), such that
and
In addition, due to the automorphism (59), there is another topological coproduct structure \(^{\sigma }\Delta =(\sigma \otimes \sigma )\circ \Delta \circ \sigma ^{-1}\), then
which means
By a direct calculation, we obtain
where
\(\rho ^\sigma _{\vec {u}}:=\prod _{k=1}^{N} \otimes \rho _{u_k}\circ {^\sigma \Delta ^{(N-1)}}\) and \(\Lambda _k^{\pm }(z)\) are given by
Note that
for any symmetric functions f and g, where the operator \(X_0^{(1)}\) is given by [56]
in which
\(X_0^{(1)}\) is the standard operator used to define the GMP [12, 13].
Let us define the polynomials \(M_{\vec {\lambda }}\) as follows:
where \({\mathcal {V}}^{\vec {\mu }}_{\vec {\lambda }}(\vec {u})\) is a complex matrix, and
in particular,
Note that our functions \(G_{\lambda ,\mu }(u)\) are identical to the functions \(N_{\lambda ,\mu }((q/t)u)\) in Ref. [12].
The dual polynomial \(M^*_{\vec {\lambda }}\{\vec {u},\vec {p}\}\) is defined by
which implies the Cauchy completeness identity
From (72) and (78), we see that \(X^{(1)}_0\) can be diagonalized by \(M^*_{\vec {\lambda }}\{\vec {u};\vec {p}\}\). Thus, \(M_{\vec {\lambda }}\{\vec {u};\vec {p}\}\) and \(M^*_{\vec {\lambda }}\{\vec {u};\vec {p}\}\) correspond to the bra and ket states of GMP, respectively [12].
According to (60a), there are a series of operators commutating with \(X_0^{+}\)
and they can be regarded as the higher Hamiltonians for GMP.
Then, we have
where the eigenvalues are given by
in which
Here we have used a conjecture on the eigenfunctions
which is equivalent to the expression conjectured by Awata et al. [74].
3.2 Limit to the \(\beta \) deformed case
In this subsection, we consider a certain limit to the \(\beta \) deformed case. Let us now consider the semi-classical limits: \(t=q^{\beta }=e^{\beta \hbar }\), \(u_k=q^{a_k}\) with \(k=1,\cdots ,N\) and taking \(\hbar \rightarrow 0\) with \(\beta \) fixed. In this case, GMP degenerate to GJP.
Comparing (38c) with (75c), we have
Thus, we may rewrite the \(\beta \)-deformed operators (33c) as
which can be regarded as the higher Hamiltonians for GJP.
Let us list some operators
The first two operators coincide with the ones derived in Ref. [75].
3.3 Generalized (q, t)-deformed partition functions
Let us construct the generalized (q, t)-deformed W-operators with the help of the recursive formulas (60b) as follows:
where \(\alpha \in \{+,-\}\), \({\mathcal {W}}^{\pm }(\vec {u};m)=\rho ^{\sigma }_{\vec {u}}({\textbf{u}}_{0,0}\mp m{\textbf{u}}_{0,\pm 1})\) and
Note that a similar approach proposed in Ref. [54] to construct (q, t)-deformed W-operators is in terms of the commutators of the W-operators and Macdonald difference operator. However its deficiency is clear. Due to the commutators, it is hard to give exact expression of the W-operators. As an improvement, our approach allows us to explicitly construct the (q, t)-deformed W-operators based on the recursive formulas (60b). That is to say that the exact expression of the W-operators can be given recursively.
In order to give the explicit actions of these operators on GMP, we introduce the generalized (q, t)-deformed cut-and-join rotation operators
such that
where
In terms of (89) and (90), the W-operators \({\mathcal {W}}^{\alpha }_{\pm l,nl}(\vec {u},\vec {m})\) can be represented as
The actions of W-operators (93) and \(W_{0,\pm n}(\vec {u};\vec {p})\) (88c) on GMP are
where the coefficient \({\mathcal {C}}_{\vec {\lambda }}^{\vec {\mu }}\) and \(\bar{{\mathcal {C}}}_{\vec {\lambda }}^{\vec {\mu }}\) are given by
With the construction of the W-operators we made above, we may give the partition functions through W-representations
where the skew polynomials \(M_{\vec {\lambda }/\vec {\mu }}\) are defined by
for any symmetric function f.
3.4 5d Nekrasov partition functions
The 5d Nekrasov partition function for the U(N) theory with \({\mathcal {N}}_f=2n\) fundamental hypermultiplets is given by [4, 5, 12]
where
The 5d Nekrasov partition function can be given by vertex operators [12, 13]. To do this, one first modifies the correspondence (44) as
where \([\hat{a}_n,\hat{a}_m]=n\frac{1-q^n}{1-t^n}\delta _{m+n,0}\).
There is the vertex operator [12]
which satisfies a Pieri-type formula
It is easy to obtain
More generally, there is the vertex operator (or Mukadé operator) \(\Phi (z)\) in [12, 13] such that
where \(\xi ^{(+)}\) is defined as a normalized coefficient in [13]. In terms of \(\Phi (z)\), it is easy to give the 5d U(N) Nekrasov partition function with \({\mathcal {N}}_{f}=2N\) fundamental matters
where \(\vec {w}^{-1}=(w_1^{-1},w_2^{-1},\ldots )\).
Going back to our partition functions (96a), we observe that by removing the generalized Jack polynomials from \({\mathcal {Z}}_{-,n}\) (96a) and taking \(m_i=(q/t)(m^+_i)^{-1}\), \(m'_i=(m^-_i)^{-1}\) with \(i=1,2,\ldots ,n\), the remains match with the 5d Nekrasov partition functions with \({\mathcal {N}}_{f}\) fundamental matters (98). With this observation, we may establish the connection between these two partition functions
Then we make the explicit connection between the W-operators (93) and vertex operator \(\Phi (z)\) from (104) and (105)
where \({\mathcal {A}}_l=\frac{1}{l}\left( \frac{q^Nz_1}{t^Nz_2}\right) ^l\frac{1-t^l}{1-q^l}\) and \(m_i=(q/t)(v_i)^{-1}\), \(m'_i=w_i\) for \(i=1,2,\ldots ,N\).
4 Conclusion
We have attempted to construct the generalized \(\beta \) and (q, t)-deformed partition functions through W-representations, where the expansions are respectively with respect to the GJP and GMP labeled by N-tuple of Young diagrams. To achieve the desired results, the key point is the construction of W-operators. Based on the N-Fock representations of the \(\mathbf {SH^c}\) and elliptic Hall algebras, we constructed generalized \(\beta \) and (q, t)-deformed W-operators, respectively. In order to present the actions of the W-operators on the GJP and GMP more explicitly, we have introduced the generalized cut-and-join rotation operators. In addition, we presented the higher Hamiltonians (80) for GMP. Furthermore, the Hamiltonians for GJP (86) can be obtained by the semi-classical limits of the operator \(X^{+}_0\) (69). Unfortunately, we failed to prove the eigenvalue conjecture (84) which is equivalent to the expression conjectured by Awata et al. [74]. We leave this as a future work.
As the particular case, based on the Fock representation (symmetric functions representation) of the algebra \(\mathbf {SH^c}\), we also constructed the \(\beta \)-deformed W-operators and presented the \(\beta \)-deformed partition functions through W-representations. It was noted that the \(\beta \)-deformed W-operators are closely related to one-dimensional many-body systems, such as the classical and U(L|M)-deformed rCS models. In addition, we found that the \(\beta \)-deformed ABJ-like model (19) is the integral realization of \(Z_{-2}(L_{eff};\bar{p})\) in (17a).
By means of the desired deformed W-operators, we have achieved the generalized \(\beta \) and (q, t)-deformed partition functions through W-representations. The most interesting result is that there are the much profound interrelations between our deformed partition functions and the 4d and 5d Nekrasov partition functions. We also noticed that the free energy of the Nekrasov partition function \(Z_{inst}^{U(1),1}(s;0;u;v)|_{\beta =1}\) (56) can be given by the specific part of (connected) Belyi fat graphs. It is well-known that the corresponding Nekrasov partition functions can be given by vertex operators. Thus the deep connection between our \(\beta \) and (q, t)-deformed W-operators and vertex operators was revealed in this paper. Owing to this remarkable connection, more applications of these deformed W-operators would merit further investigations. For further research, it would also be interesting to search for the integral representations for the generalized \(\beta \) and (q, t)-deformed partition functions.
Data Availability Statement
This manuscript has no associated data. [Authors’ comment: This is a theoretical study, and no experimental data has been listed.]
Code Availability Statement
The manuscript has no associated code/software. [Author’s comment: This is a theoretical study, and no associated code/software is involved.]
References
N.A. Nekrasov, Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831 (2004). arXiv:hep-th/0206161
N. Nekrasov, A. Okounkov, Seiberg–Witten theory and random partitions. Prog. Math. 244, 525 (2006). arXiv:hep-th/0306238
L.F. Alday, D. Gaiotto, Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167 (2010). arXiv:0906.3219
H. Awata, Y. Yamada, Five-dimensional AGT relation and the deformed \(\beta \)-ensemble. Prog. Theor. Phys. 124, 227 (2010). arXiv:1004.5122
M.C. Tan, An M-theoretic derivation of a \(5d\) and \(6d\) AGT correspondence, and relativistic and elliptized integrable systems. JHEP 12, 031 (2013). arXiv:1309.4775
N. Wyllard, Instanton partition functions in \(N=2\)\(SU(N)\) gauge theories with a general surface operator and their \(W\)-algebra duals. JHEP 02, 114 (2011). arXiv:1012.1355
V.A. Alba, V.A. Fateev, A.V. Litvinov, G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture. Lett. Math. Phys. 98, 33 (2011). arXiv:1012.1312
V.A. Fateev, A.V. Litvinov, Integrable structure, \(W\)-symmetry and AGT relation. JHEP 01, 051 (2012). arXiv:1109.4042
S. Kanno, Y. Matsuo, H. Zhang, Extended conformal symmetry and recursion formulae for Nekrasov partition function. JHEP 08, 028 (2013). arXiv:1306.1523
Y. Matsuo, C. Rim, H. Zhang, Construction of Gaiotto states with fundamental multiplets through degenerate DAHA. JHEP 09, 028 (2014). arXiv:1405.3141
J.E. Bourgine, M. Fukuda, Y. Matsuo, H. Zhang, R.D. Zhu, Coherent states in quantum \(W_{1+\infty }\) algebra and \(qq\)-character for 5\(d\) super Yang–Mills. Prog. Theor. Exp. Phys. 2016, 123B05 (2016). arXiv:1606.08020
H. Awata, B. Feigin, A. Hoshino, M. Kanai, J. Shiraishi, S. Yanagida, Notes on Ding-Iohara algebra and AGT conjecture. RIMS Kokyuroku 1765, 12 (2011). arXiv:1106.4088
M. Fukuda, Y. Ohkubo, J. Shiraishi, Generalized Macdonald functions on Fock tensor spaces and duality formula for changing preferred direction. Commun. Math. Phys. 380, 1 (2020). arXiv:1903.05905
A. Smirnov, Polynomials associated with fixed points on the instanton moduli space. arXiv:1404.5304
A. Tsymbaliuk, The affine Yangian of \(\mathfrak{gl} _1\) revisited. Adv. Math. 304, 583 (2017). arXiv:1404.5240
O. Schiffmann, E. Vasserot, Cherednik algebras, \(W\)-algebras and the equivariant cohomology of the moduli space of instantons on \({\mathbb{A} }^2\). Publ. Math. Inst. Hautes Études Sci. 118, 213 (2013). arXiv:1202.2756
B. Feigin, E. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Quantum toroida \(\mathfrak{gl} _{1}\)-algebra: plane partitions. Kyoto J. Math. 52, 621 (2012). arXiv:1110.5310
V. Dotsenko, V. Fateev, Conformal algebra and multipoint correlation functions in two-dimensional statistical models. Nucl. Phys. B 240, 312 (1984)
A. Mironov, A. Morozov, Sh. Shakirov, A direct proof of AGT conjecture at \(\beta =1\). JHEP 02, 067 (2011). arXiv:1012.3137
A. Morozov, A. Smirnov, Towards the proof of AGT relations with the help of the generalized Jack polynomials. Lett. Math. Phys. 104, 585 (2014). arXiv:1307.2576
S. Mironov, A. Morozov, Y. Zenkevich, Generalized Jack polynomials and the AGT relations for the \(SU(3)\) group. JETP Lett. 99, 115 (2014). arXiv:1312.5732
Y. Zenkevich, Generalized Macdonald polynomials, spectral duality for conformal blocks and AGT correspondence in five dimensions. JHEP 05, 131 (2015). arXiv:1412.8592
A. Morozov, Y. Zenkevich, Decomposing Nekrasov decomposition. JHEP 02, 098 (2016). arXiv:1510.01896
H. Zhang, Y. Matsuo, Selberg integral and \(SU(N)\) AGT conjecture. JHEP 12, 106 (2011). arXiv:1110.5255
H. Itoyama, K. Maruyoshi, T. Oota, The quiver matrix model and 2\(d\)-4\(d\) conformal connection. Prog. Theor. Phys. 123, 957 (2010). arXiv:0911.4244
A. Mironov, A. Morozov, S. Shakirov, Towards a proof of AGT conjecture by methods of matrix models. Int. J. Mod. Phys. A 27, 1230001 (2012). arXiv:1011.5629
T. Nishinaka, C. Rim, \(\beta \)-deformed matrix model and Nekrasov partition function. JHEP 02, 114 (2012). arXiv:1112.3545
T. Tai, Instanton counting and matrix model. Prog. Theor. Phys. 119, 165 (2008). arXiv:0709.0432
P. Sulkowski, Matrix models for \(\beta \)-ensembles from Nekrasov partition functions. JHEP 04, 063 (2010). arXiv:0912.5476
G. Bonelli, K. Maruyoshi, A. Tanzini, F. Yagi, Generalized matrix models and AGT correspondence at all genera. JHEP 07, 055 (2011)
A. Mironov, A. Morozov, Superintegrability as the hidden origin of the Nekrasov calculus. Phys. Rev. D 106, 126004 (2022). arXiv:2207.08242
A. Morozov, Sh. Shakirov, Generation of matrix models by \(\hat{W}\)-operators. JHEP 04, 064 (2009). arXiv:0902.2627
A. Alexandrov, Cut-and-Join operator representation for Kontsewich–Witten tau-function. Mod. Phys. Lett. A 26, 2193 (2011). arXiv:1009.4887
A. Alexandrov, Cut-and-join description of generalized Brezin–Gross–Witten model. Adv. Theor. Math. Phys. 22, 1347 (2018). arXiv:1608.01627
L. Cassia, R. Lodin, M. Zabzine, On matrix models and their \(q\)-deformations. JHEP 10, 126 (2020). arXiv:2007.10354
A. Mironov, V. Mishnyakov, A. Morozov, Non-Abelian \(W\)-representation for GKM. Phys. Lett. B 823, 136721 (2021). arXiv:2107.02210
I. Goulden, D. Jackson, R. Vakil, The Gromov–Witten potential of a point, Hurwitz numbers, and Hodge integral. Proc. Lond. Math. Soc. 83, 563 (2001). arXiv:math/9910004
A. Alexandrov, A. Mironov, A. Morozov, S. Natanzon, On KP-integrable Hurwitz functions. JHEP 11, 080 (2014). arXiv:1405.1395
R. Wang, F. Liu, C.H. Zhang, W.Z. Zhao, Superintegrability for (\(\beta \)-deformed) partition function hierarchies with \(W\)-representations. Eur. Phys. J. C 82, 902 (2022). arXiv:2206.13038
A. Alexandrov, On \(W\)-operators and superintegrability for dessins d’enfant. Eur. Phys. J. C 83, 147 (2023). arXiv:2212.10952
A. Mironov, V. Mishnyakov, A. Morozov, A. Popolitov, W.Z. Zhao, On KP-integrable skew Hurwitz \(\tau \)-functions and their \(\beta \)-deformations. Phys. Lett. B 839, 137805 (2023). arXiv:2301.11877
A. Mironov, V. Mishnyakov, A. Morozov, A. Popolitov, R. Wang, W.Z. Zhao, Interpolating matrix models for WLZZ series. Eur. Phys. J. C 83, 377 (2023). arXiv:2301.04107
A. Mironov, A. Oreshina, A. Popolitov, Two \(\beta \)-ensemble realization of \(\beta \)-deformed WLZZ models. arXiv:2403.05965
A. Mironov, A. Oreshina, A. Popolitov, \(\beta \)-WLZZ models from \(\beta \)-ensemble integrals direactly. arXiv:2404.18843
A. Mironov, V. Mishnyakov, A. Morozov, A. Popolitov, Commutative families in \(W_{\infty }\), integrable many-body systems and hypergeometric \(\tau \)-functions. JHEP 09, 065 (2023). arXiv:2306.06623
R.J. Szabo, M. Tierz, \(q\)-deformations of two-dimensional Yang–Mills theory: classification, categorification and refinement. Nucl. Phys. B 876, 234 (2013). arXiv:1305.1580
Z. Kökényesi, A. Sinkovics, R.J. Szabo, Refined Chern–Simons theory and (\(q, t\))-deformed Yang–Mills theory: semi-classical expansion and planar limit. JHEP 10, 067 (2013). arXiv:1306.1707
H. Awata, H. Kanno, T. Matsumoto, A. Mironov, A. Morozov, A. Morozov, Y. Ohkubo, Y. Zenkevich, Explicit examples of DIM constraints for network matrix models. JHEP 07, 103 (2016). arXiv:1604.08366
A. Mironov, A. Morozov, Y. Zenkevich, Ding–Iohara–Miki symmetry of network matrix models. Phys. Lett. B 762, 196 (2016). arXiv:1603.05467
A. Morozov, A. Popolitov, Sh. Shakirov, On (\(q, t\))-deformation of Gaussian matrix model. Phys. Lett. B 784, 342 (2018). arXiv:1803.11401
A. Morozov, On \(W\)-representations of \(\beta \)- and \(q, t\)-deformed matrix models. Phys. Lett. B 792, 205 (2019). arXiv:1901.02811
A. Morozov, A. Mironov, Elliptic \(q, t\) matrix models. Phys. Lett. B 816, 136196 (2021). arXiv:2011.01762
A. Morozov, A. Mironov, Towards elliptic deformation of \(q, t\) matrix models. Phys. Lett. B 816, 136221 (2021). arXiv:2011.02855
F. Liu, A. Mironov, V. Mishnyakov, A. Morozov, A. Popolitov, R. Wang, W.Z. Zhao, \((q, t)\)-deformed (skew) Hurwitz \(\tau \)-functions. Nucl. Phys. B 993, 116283 (2023). arXiv:2303.00552
J.E. Bourgine, A. Garbali, A \((q,t)\)-deformation of the 2\(d\) Toda integrable hierarchy. arXiv:2308.16583
B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, S. Yanagida, A commutative algebra on degenerate \(\mathbb{C}\mathbb{P} ^{1}\) and Macdonald polynomials. J. Math. Phys. 50, 095215 (2009). arXiv:0904.2291
J. Ding, K. Iohara, Generalization and deformation of Drinfeld quantum affine algebras. Lett. Math. Phys. 41, 181 (1997). arXiv:q-alg/9608002
K. Miki, A \((q,\gamma )\) analog of the \(W_{1+\infty }\) algebra. J. Math. Phys. 48, 3520 (2007)
B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Finite type modules and Bethe ansatz for quantum toroidal \(\mathfrak{gl} _1\). Commun. Math. Phys. 356, 285 (2017). arXiv:1603.02765
J.E. Bourgine, M. Fukuda, K. Harada, Y. Matsuo, R.D. Zhu, \((p, q)\)-webs of DIM representations, \(5d\)\(N=1\) instanton partition functions and \(qq\)-characters. JHEP 11, 034 (2017). arXiv:1703.10759
H. Awata, H. Fujino, Y. Ohkubo, Crystallization of deformed Virasoro algebra, Ding–Iohara–Miki algebra, and \(5D\) AGT correspondence. J. Math. Phys. 58, 071704 (2017)
I. Burban, O. Schiffmann, On the Hall algebra of an elliptic curve, I. Duke Math. J. 161, 1171 (2012). arXiv:math/0505148
O. Schiffmann, E. Vasserot, The elliptic Hall algebra and the K-theory of the Hilbert scheme of \({\mathbb{A} }^2\). Duke. Math. J. 162, 279 (2013). arXiv:0905.2555
T. Procházka, \(W\)-symmetry, topological vertex and affine Yangian. JHEP 10, 077 (2016). arXiv:1512.07178
H. Nakajima, More lectures on Hilbert schemes of points on surfaces. arXiv:1401.6782
M. Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces. Invent. Math. 136, 157 (1999). arXiv:math/9803091
A.N. Sergeev, A.P. Veselov, Generalised discriminants, deformed Calogero–Moser-Sutherland operators and super-Jack polynomials. Adv. Math. 192, 341 (2005). arXiv:math-ph/0307036
I.G. Macdonald, Symmetric Functions and Hall Polynomials (Oxford Science Publications, Oxdford, 1995)
F.H. Zhang, F. Liu, Y. Li, C.H. Zhang, On higher deformed Calogero–Sutherland Hamiltonians (Submitted)
L. Cassia, R. Lodin, M. Zabzine, Virasoro constraints revisited. Commun. Math. Phys. 387, 1729 (2021). [arXiv:2102.05682]
E. Carlsson, A. Okounkov, Exts and vertex operators. Duke Math. J. 161, 1797 (2012). [arXiv:0801.2565]
J. Zhou, Grothendieck’s dessins d’enfants in a web of dualities. arXiv:1905.10773
Y. Zenkevich, On pentagon identity in Ding–Iohara–Miki algebra. JHEP 03, 193 (2023). [arXiv:2112.14687]
H. Awata, H. Kanno, A. Mironov, A. Morozov, A. Morozov, Y. Ohkubo, Y. Zenkevich, Toric Calabi-Yau threefolds as quantum integrable systems. \(R\)-matrix and \(RTT\) relations. JHEP 10, 047 (2016). arXiv:1608.05351
Y. Ohkubo, Generalized Jack and Macdonald polynomials arising from AGT conjecture. J. Phys. Conf. Ser. 804, 012036 (2017). arXiv:1404.5401
Funding
This work is supported by the National Natural Science Foundation of China (nos. 12375004 and 12205368) and the Fundamental Research Funds for the Central Universities, China (no. 2024ZKPYLX01).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funded by SCOAP3.
About this article
Cite this article
Liu, F., Wang, R., Yang, J. et al. Generalized \(\beta \) and (q, t)-deformed partition functions with W-representations and Nekrasov partition functions. Eur. Phys. J. C 84, 756 (2024). https://doi.org/10.1140/epjc/s10052-024-13040-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjc/s10052-024-13040-w