Discrete Painlevé system for the partition function of N _f  =  2 SU(2) supersymmetric gauge theory and its double scaling limit

Abstract

We continue to study the matrix model of the N f = 2 SU (2) case that represents the irregular conformal block. What provides us with the Painlevé system is not the instanton partition function per se but rather a finite analog of its Fourier transform that can serve as a generating function. The system reduces to the extension of the Gross-Witten-Wadia unitary one-matrix model by the logarithmic potential while keeping the planar critical behavior intact. The double scaling limit to this critical point is a constructive way to study Argyres-Douglas type theory from IR. We elaborate upon the method of orthogonal polynomials and its relevance to these problems, developing it further for the case of a generic unitary matrix model and that of a special case with the logarithmic potential. *

Argyres-Douglas theories, S-duality and AGT correspondence

“…While the partition functions of conformally gauged AD theories have not been evaluated, there exists a series of non-conformally gauged AD theories whose partition functions were evaluated via a generalization [17,18] of the AGT correspondence [19,20] (See [21][22][23][24][25][26][27][28][29][30] for recent developments on this generalization). In particular, for the theory described by the quiver in figure 2, the partition function was evaluated as the inner product of so-called "irregular states" of Virasoro algebra.…”

Argyres-Douglas theories and Liouville irregular states

“…Indeed, an attempt to compute the partition function of AD theories as Liouville irregular conformal blocks was partially studied in [30,31]. 4 Finally, nice matrix model descriptions of this relation between AD theories and Painlevé equations have been studied in [37][38][39]. 5 In this paper, we extend the irregular conformal block approach to various directions, focusing on the (A 1 , A 3 ) and (A 1 , D 4 ) AD theories.…”

Argyres-Douglas Theories, S-duality and AGT Correspondence

“…Indeed, it was shown in [5] that P SL(2, Z) naturally acts on a modified electro-magnetic charge lattice where the "electric charge" is such a linear combination of charges arising from the three sectors. 26 This linear combination naturally arises when constructing the (A 3 , A 3 ) theory as the IR limit of type IIB string theory on a Calabi-Yau singularity. With this modified charge lattice, the minimal electric charge is now half the charge of fundamental quark, 27 which is consistent with our T-transformation (6.2).…”

“…We list some of them below. 26 The "magnetic charge" here is simply associated with the SU (2) vector multiplet without mixing with those arising from the (A 1 , D 4 ) sectors. 27 This can be seen from the discussions around Eq.…”

“…While the partition functions of conformally gauged AD theories have not been evaluated, there exists a series of non-conformally gauged AD theories whose partition functions were evaluated via a generalization [14,15] of the AGT correspondence [16,17] (See [18][19][20][21][22][23][24][25][26][27] (A 1 , D 2n )…”