$\mathcal{N}=2^{\ast }$ 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 $SU(2)$ $\mathcal{N}=2^{\ast }$ 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 $S{L}_2$ 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 $SU(2)$ $\mathcal{N}=2^{\ast }$ theory on self-dual $\Omega$-background and, in the Seiberg-Witten limit, an elegant relation between the IR and UV gauge couplings.

arXiv comment40 pages, 6 figures

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• Non-perturbative approaches to the quantum Seiberg-Witten curve
• A slow review of the AGT correspondence
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• Elliptic Blowup Equations for 6d SCFTs. III: E-strings, M-strings and Chains
• Intersecting Defects and Supergroup Gauge Theory
• BPS quivers of five-dimensional SCFTs, Topological Strings and q-Painlevé equations
• Quantum spectral problems and isomonodromic deformations
• Instantons to the people: the power of one-form symmetries
• Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions
• Four flavours, triality and bimodular forms
• $t{t}^{\ast }$ Toda equations for surface defects in $\mathcal{N}=2$ SYM and instanton counting for classical Lie groups
• Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus
• On a 5D UV completion of Argyres-Douglas theories
• Painlevé/CFT correspondence on a torus
• AGT correspondence, (q-)Painlevè equations and matrix models