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arXiv · 1901.10497

$\mathcal{N}=2^*$ gauge theory, free fermions on the torus and Painlev\'e VI

Giulio Bonelli·Fabrizio Del Monte·Pavlo Gavrylenko·Alessandro Tanzini
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Abstract

In this paper we study the extension of Painlev\'e/gauge theory correspondence to circular quivers by focusing on the special case of $SU(2)$ $\mathcal{N}=2^*$ 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 $SU(2)$ $\mathcal{N}=2^*$ theory on self-dual $\Omega$-background and, in the Seiberg-Witten limit, an elegant relation between the IR and UV gauge couplings.

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Giulio Bonelli, Fabrizio Del Monte, Pavlo Gavrylenko, Alessandro Tanzini. 2019-01-29. $\mathcal{N}=2^*$ gauge theory, free fermions on the torus and Painlev\'e VI. https://doi.org/10.1007/s00220-020-03743-y

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