Opers and TBA

TL;DR

This paper explores the conformal limit of TBA equations related to N=2 gauge theories, revealing a connection to opers and solutions to the Schrödinger equation with rational potentials.

Contribution

It introduces a generalized oper submanifold in the space of flat connections via conformal TBA equations, linking gauge theory, geometry, and quantum mechanics.

Findings

Conformal TBA equations describe a generalized oper submanifold.

Solutions to Schrödinger equations with rational potentials are obtained.

The work connects gauge theory moduli spaces with quantum differential equations.

Abstract

In this note we study the "conformal limit" of the TBA equations which describe the geometry of the moduli space of four-dimensional N=2 gauge theories compactified on a circle. We argue that the resulting conformal TBA equations describe a generalization of the oper submanifold in the space of complex flat connections on a Riemann surface. In particular, the conformal TBA equations for theories in the A1 class produce solutions of the Schr\"odinger equation with a rational potential.