Non-perturbative approaches to the quantum Seiberg-Witten curve

TL;DR

This paper explores non-perturbative quantization methods for the Seiberg-Witten curve in N=2 SU(2) super Yang-Mills theory, linking quantum WKB, TBA equations, instanton calculus, and Painleve III tau functions.

Contribution

It introduces a unified framework connecting quantum WKB, TBA equations, and the TS/ST correspondence to analyze the quantum Seiberg-Witten curve.

Findings

Quantum WKB periods exhibit resurgent properties linked to TBA equations.

Borel-resummed quantum periods relate to instanton calculus.

Closed formula for the Fredholm determinant of the modified Mathieu operator obtained.

Abstract

We study various non-perturbative approaches to the quantization of the Seiberg-Witten curve of ${\cal N}=2$, $SU(2)$ super Yang-Mills theory, which is closely related to the modified Mathieu operator. The first approach is based on the quantum WKB periods and their resurgent properties. We show that these properties are encoded in the TBA equations of Gaiotto-Moore-Neitzke determined by the BPS spectrum of the theory, and we relate the Borel-resummed quantum periods to instanton calculus. In addition, we use the TS/ST correspondence to obtain a closed formula for the Fredholm determinant of the modified Mathieu operator. Finally, by using blowup equations, we explain the connection between this operator and the $\tau$ function of Painleve $\rm III$.