N=2 supersymmetric QCD and elliptic potentials

TL;DR

This paper explores the deep connection between N=2 supersymmetric gauge theories and elliptic quantum mechanics, providing a detailed analysis of spectral properties and their relation to gauge theory instanton effects.

Contribution

It establishes a precise correspondence between the spectrum of elliptic quantum mechanics and the instanton partition function in N=2 SU(2) super Yang-Mills theory, including asymptotic expansions.

Findings

Derived asymptotic spectrum expansions for the Schrödinger operator.

Fixed the relation between energy spectrum and gauge theory instanton partition function.

Analyzed strong coupling spectral behavior in Seiberg-Witten theory.

Abstract

We investigate the relation between the four dimensional N=2 SU(2) super Yang-Mills theory with four fundamental flavors and the quantum mechanics model with Treibich-Verdier potential described by the Heun equation in the elliptic form. We study the precise correspondence of quantities in the gauge theory and the quantum mechanics model. An iterative method is used to obtain the asymptotic expansion of the spectrum for the Schr\"{o}dinger operator, we are able to fix the precise relation between the energy spectrum and the instanton partition function of the gauge theory. We also study asymptotic expansions for the spectrum which correspond to the strong coupling regions of the Seiberg-Witten theory.