Matrix model from $ \mathcal{N} = 2 $ orbifold partition function

Abstract

The orbifold generalization of the partition function, which would describe the gauge theory on the ALE space, is investigated from the combinatorial perspective. It is shown that the root of unity limit q → exp(2πi/k) of the q-deformed partition function plays a crucial role in the orbifold projection while the limit q → 1 applies to R 4 . Then starting from the combinatorial representation of the partition function, a new type of multimatrix model is derived by considering its asymptotic behavior. It is also shown that Seiberg-Witten curve for the corresponding gauge theory arises from the spectral curve of this multi-matrix model. *

Fractional quiver W-algebras

“…This construction is similar, but different from the instanton counting on the orbifold (C/Z d i ) × C itself, which is used to implement the surface operator [23][24][25]. See also [32,33] for a realization of the orbifold using the equivariant parameter.…”

4d index to 3d index and 2d topological quantum field theory

“…When the lens space is a three sphere, the 2d theory is proposed to be the q-deformed Yang-Mills theory [19]. To identify the 2d counterpart of the orbifold index, it would be important to understand its relation to the AGT correspondence [30] between 4d N = 2 partition functions on ALE spaces and 2d Para-Liouville/Toda theories [31,32,33,34,35] (see also [36,37,38]). In a similar way, one could consider the 6d N = (2, 0) A 1 theory compactified on a Riemann surface with N = 1 twist, giving rise to 4d N = 1 theories [39].…”

Orbifolds and Exact Solutions of Strongly-Coupled Matrix Models

“…An important special case that has attracted significant attention is when the q-deformation parameter is an rth root of unity. In this case, as shown in [20,64], the 5d partition function reduces to a 4d partition function on an ALE space, the C 2 /Z r orbifold. At the same time the matrix model (q-Selberg integral) reduces to precisely the integrals that we study in the current paper, see §4.3.…”

“…Finally, another possible avenue of application for our results is described in §5.2 concerns a generalization of the AGT conjecture [18,19]. In [20] the partition function of four-dimensional gauge theory on an orbifold C 2 /Z r was studied resulting in matrix models similar to those considered here. It would be interesting to determine the precise connection, and to understand if our results may be used to compute three-point functions in para-Liouville theory (see, e.g., [21][22][23][24][25]), thus generalizing [26].…”