# Spectra of elliptic potentials and supersymmetric gauge theories

## TL;DR

This paper explores the deep connection between the spectral properties of a complex elliptic potential in quantum mechanics and the non-perturbative aspects of supersymmetric gauge theories, revealing dualities and exact relations.

## Contribution

It establishes a novel link between the spectra of the Darboux-Treibich-Verdier elliptic potential and the instanton partition functions in supersymmetric QCD, including duality relations.

## Key findings

- Weak coupling spectral solutions relate to instanton partition functions.
- Strong coupling solutions correspond to gauge theory prepotential expansions.
- Duality transformations connect different spectral and gauge theory regimes.

## Abstract

We study a relation between asymptotic spectra of the quantum mechanics problem with a four components elliptic function potential, the Darboux-Treibich-Verdier (DTV) potential, and the Omega background deformed N=2 supersymmetric SU(2) QCD models with four massive flavors in the Nekrasov-Shatashvili limit. The weak coupling spectral solution of the DTV potential is related to the instanton partition function of supersymmetric QCD with surface operator. There are two strong coupling spectral solutions of the DTV potential, they are related to strong coupling expansions of gauge theory prepotential at the magnetic and dyonic points in the moduli space. A set of duality transformations relate the two strong coupling expansions for spectral solution, and for gauge theory prepotential.