Exact WKB methods in $SU(2)$ $N_f=1$

TL;DR

This paper investigates the exact WKB analysis of a Schrödinger equation linked to $SU(2)$ $N_f=1$ SQCD, computing quantum periods and spectra through multiple methods, and exploring the connection between Borel singularities and BPS states.

Contribution

It provides a comprehensive comparison of different methods for calculating quantum periods and spectra in the $SU(2)$ $N_f=1$ theory, and elucidates the relationship between Borel singularities and BPS states.

Findings

Good agreement among all computational methods for quantum periods and spectra.

Identification of Borel singularities with BPS states in 4d and 2d+4d systems.

Explicit computation of the Fredholm determinant using TBA and TS/ST correspondence.

Abstract

We study in detail the Schr\"{o}dinger equation corresponding to the four dimensional $SU(2)$ $\mathcal{N}=2$ SQCD theory with one flavour. We calculate the Voros symbols, or quantum periods, in four different ways: Borel summation of the WKB series, direct computation of Wronskians of exponentially decaying solutions, the TBA equations of Gaiotto-Moore-Neitzke/Gaiotto, and instanton counting. We make computations by all of these methods, finding good agreement. We also study the exact quantization condition for the spectrum, and we compute the Fredholm determinant of the inverse of the Schr\"odinger operator using the TS/ST correspondence and Zamolodchikov's TBA, again finding good agreement. In addition, we explore two aspects of the relationship between singularities of the Borel transformed WKB series and BPS states: BPS states of the 4d theory are related to singularities in the Borel transformed WKB series for the quantum periods, and BPS states of a coupled 2d+4d system are related to singularities in the Borel transformed WKB series for local solutions of the Schr\"odinger equation.