$${\mathcal {N}}$$ = $$2^*$$ Gauge Theory, Free Fermions on the Torus and Painlevé VI

Abstract

In this paper we study the extension of Painlevé/gauge theory correspondence to circular quivers by focusing on the special case of SU (2) N = 2 * theory. We show that the Nekrasov-Okounkov partition function of this gauge theory provides an explicit combinatorial expression and a Fredholm determinant formula for the tau-function describing isomonodromic deformations of SL 2 flat connections on the one-punctured torus. This is achieved by reformulating the Riemann-Hilbert problem associated to the latter in terms of chiral conformal blocks of a free-fermionic algebra. This viewpoint provides the exact solution of the renormalization group flow of the SU (2) N = 2 * theory on self-dual Ωbackground and, in the Seiberg-Witten limit, an elegant relation between the IR and UV gauge couplings.

BPS Quivers of Five-Dimensional SCFTs, Topological Strings and q-Painlevé Equations

“…In the four-dimensional case, the Painlevé/gauge theory correspondence [42] extends also to non-toric cases, corresponding to isomonodromic deformation problems on higher genus Riemann surfaces, see [71,72] for the genus one case. These have a 5d uplift in terms of q-Virasoro algebra [73] and matrix models [74,75] whose BPS quiver interpretation would be more than welcome.…”

Non-perturbative approaches to the quantum Seiberg-Witten curve

“…Heun operator [103] (see also [104]). Likewise the spectrum of the Calogero-Moser system should make contact with the τ function describing the isomonodromic deformations on the torus [105]. The details will appear somewhere else [106].…”

Circular quiver gauge theories, isomonodromic deformations and $$W_N$$ fermions on the torus

“…In this paper, we will analyze how the identification between gauge theory partition function and the τ -function of a suitable isomomonodromy deformation problem (of which Painlevé equations constitute the simplest instance) arises for a A N −1 class S theories on the torus, a typical example of which is a circular quiver N = 2 d = 4 SU (N ) SUSY gauge theory, depicted in Fig. 1, in a self-dual -background and which are the integrable systems involved, generalizing the result of [7], where the simplest Fig. 1 Circular quiver gauge theory corresponding to the torus with n punctures: for every puncture z i we have a hypermultiplet of mass m i sitting in the bifundamental representation of two different SU (N ) gauge groups.…”