VEV of Baxter's Q-operator in N=2 gauge theory and the BPZ differential equation

In this short notes using AGT correspondence we express simplest fully degenerate primary fields of Toda field theory in terms an analogue of Baxter's $Q$-operator naturally emerging in ${\cal N}=2$ gauge theory side. This quantity can be considered as a generating function of simple trace chiral operators constructed from the scalars of the ${\cal N}=2$ vector multiplets. In the special case of Liouville theory, exploring the second order differential equation satisfied by conformal blocks including a degenerate at the second level primary field (BPZ equation) we derive a mixed difference-differential relation for $Q$-operator. Thus we generalize the $T$-$Q$ difference equation known in Nekrasov-Shatashvili limit of the $\Omega$-background to the generic case.

In [31] one of present authors (R.P.) has investigated the link between Deformed Seiberg-Witten curve equation and underlying Baxter's T − Q equation in gauge theory side and the null-vector decoupling equation [32] of 2d CFT in quite general setting of linear quiver gauge theories with U (n) gauge groups and 2d A n−1 Toda field theory multi-point conformal blocks in semi-classical limit (see also [33] for an earlier discussion on the role of degenerate fields in AGT correspondence).
In this short notes we'll extend some of the results of [31] to the case of generic Ω-background corresponding to the genuine quantum conformal blocks. For technical reasons we'll restrict ourselves to the case of U (2) gauge groups corresponding to the Liouville theory leaving Toda field theory case for future work.
In Section 2 we show that an appropriate choice of parameters [34] in A r+1 linear quiver theory with U (n) gauge groups is equivalent to insertion of the analoge of Baxters Q-operator into the partition function of a theory with one gauge node less A r theory with generic parameters. In the 2d CFT side such special choice corresponds to insertion of a degenerated primary field in the conformal block [34]. In Section 2 restricting to the case of Liouville theory, starting from the second order differential equation satisfied by the multi-points conformal blocks including a degenerate field V −b/2 [32] we derive the analogues equation satisfied by the gauge theory partition function with Q operator insertion. Then we show that this equation leads to a mixed linear difference-differential equation for Q operators which is a direct generalization of the T − Q equation from NS limit to the case of generic Ω-Background. Finally we summarize our results and discuss a couple of further directions which we think are worth pursuing.

A special choice of parameters, leading to Q Y insertion
Consider the instanton partition function of the linear quiver theory A r+1 with gauge groups U (n) with parameters specified as in Fig.1a. Note that the parameters of the first gauge factor (depicted as a dashed circle) are chosen to beã 0,u = a 0,u − 1 δ 1,u , where a 0,u are the parameters of the "frozen" node corresponding to the n antifundamental hypermultiplets. It has been shown in [34] that under such choice of parameters all n-tuples of Young diagrams Y0 ,u corresponding to the special node0 (the dashed circle) give no contribution in partition function unless the first diagram Y0 ,1 consists of a single column while the remaining n−1 diagrams are empty. Taking into account this huge simplification we'll be able to separate the contribution of the special node explicitly. According to the rules of construction of the partition function for this contribution we have n u,v=1 where for a pair of Young diagrams λ, µ the bifundamental contribution is given by where the arm and leg lengths of a box s A λ (s) and L λ (s) towards a Young diagram λ are defined as where (i, j) are coordinates of the box s with respect to the center of the corner box and λ i (λ j ) is the i-th column length (j-th row length) of λ as shown in Fig.2.
Using (2.2) It is not difficult to compute the factors Z bf present in (2.1). In particular To present the final result for the contribution (2.1) it is convenient to introduce the notation The analogues quantity was instrumental in construction of Baxters T-Q relation in the context of Nekrasov-Shatashvili limit of N = 2 gauge theories [13]. Recently the importance of this quantity in the case generic Ω-background was emphasized in [35]. A careful examination shows that the contribution (2.1) can be conveniently represented as is the Pochammer's symbol. Using (2.4) we can see that the Young diagram dependent part of factor Q in the denominator can be absorbed in the double product. The net effect is a simple replacement of parametersã u,0 by a u,0 in arguments of the functions Z bf : Thus we conclude that k-instanton sector of the dashed circle in A r+1 linear quiver theory can be treated as insertion of the operator in a generic A r theory. It was already known [34], that the special choice of parametersã 0,u = a 0,u − 1 δ u,1 corresponds to the insertion of the completely degenerate field V −bω 1 (z) in AGT dual Toda CFT conformal block. Thus (2.10) gives an explicit realization of this field in terms of N = 2 gauge theory notions. Until now we were discussing arbitrary gauge U (n) gauge factors. In what follows, we'll restrict ourselves with the case n = 2, corresponding to the Liouville theory in AGT dual side. The reason is that in Liouville theory conformal blocks including this degenerate field, satisfy second order differential equation 1 . In remaining part of the paper we'll translate this differential equation in gauge theory terms, finding a linear difference-differential equation, satisfied by the expectation values of the operators Q(v). Since the equation is valid for infinitely many discrete values of the spectral parameter v = a 0,1 + k 2 , k = 0, 1, 2, . . ., it can be argued that it is valid for generic values of v as well. The last statement we have checked also by explicit low order instanton computations.

Degenerate field decoupling equation in Liouville theory
Let us briefly remind that the Liouvill theory (see e.g. [36]) is characterized by the central charge c of Virasoro algebra parameterized as where b is the Liouvill's dimensionless coupling constant related to the Ω-background parameters via The conformal dimensions of primary fields are V λ are given by The parameters α are usually referred as charges. One alternatively uses the Liouville momenta P = Q/2 − λ. In Fig.1b we found it convenient to specify the fields associated to the horizontal lines by their momenta, while those of vertical lines by charges. The relations between this parameters and the gauge theory VEV's are very simple 2 for α = 2, 3, . . . , r + 1. With the same logic we have Notice that the field V λ0 = V −b/2 is indeed a degenerate field satisfying second order differential equation due to the null vector decoupling condition (below L m are the Virasoro generators) The differential equation satisfied by our r + 4-point conformal block h(λ α ) and λ r+2 = Q/2 − p r+1 . (3.9) According to AGT correspondence the instanton part of the partition function of the N = 2 theory considered in previous section with U (2) gauge group factors is related to the conformal block (3.7) as To complete the map (3.4), (3.4) between two sides let us mention also that the exponentiated gauge couplings (instanton counting parameters) are related to the insertion points as [22] q α = z α+1 /z α ; for α = 1, . . . , r , (3.11) the remaining coupling associated to the special node0 is just 1/z and z 1 = 1. In (3.10) besides standard AGT U (1) factors an extra power of z responsible for scale transformation (with scaling factor z) mapping the insertion points shown in Fig.1b to those of the conformal block (3.7). Inserting (3.10) into (3.8) and replacing CFT parameters by their gauge theory counterparts we'll find a differential equation satisfied by the partition function. After tedious but straightforward transformations it is possible to represent this equation as (for more details on calculations of this kind see [31]) r+1 α=0 (−) α χ α (− 2 z∂ z ;û 1 , . . . ,û r+1 )z −α−a 0,1 / 2 Z inst = 0 (3.12) whereû 1 = − 1 2 r+1 α=2 z α ∂ zα ;û α = 1 2 z α ∂ zα for α = 2, . . . , r + 1 (3.13) and χ α (v; u 1 , . . . , u r+1 ) are quadratic in v and linear in u 1 , . . . , u r+1 polynomials (we use notation = 1 + 2 ) (3.14) where for α = 0, 1, . . . , r + 1 We set by definition and for the other extreme value α = r + 1 it is easy to see that Representing Z inst as a power series in 1/z , from eq. (3.12) for the coefficients Q(v) we get the relation which is valid for infinitely many values v ∈ a 0,1 + 2 Z. Since Z inst is regular at z = ∞, in fact we have nontrivial equations only for v k = a 0,1 + 2 k, with k ≥ 0. Remind now that as discussed in previous section, due to eqs. (2.9), (2.10), Z inst of the A r+1 theory up to a simple factor is the same as VEV of the quantity Q Y 1 (2.10) calculated in the framework of A r gauge theory (i.e. in theory without the dashed circle in Fig.1a). Explicitly where the constant C takes the value

Summary
Thus we made an explicit link between the insertion of the Q operator in N = 2 gauge theory and insertion of simplest degenerate field in AGT dual 2d CFT.
In the special case of the gauge groups U (2) we found analog of the Baxter's T − Q equation, previously known only in the Nekrasov-Shatashvili limit of the Ω-background [13][14][15][16][17].
To conclude let us mention that a "microscopic" proof of this statement e.g. along the line presented in [37] to prove qq-character identities of [35] would be highly desirable.
Another important contribution would be generalization of our analysis to the case of arbitrary U (n) or other choices of gauge groups.