TBA for non-perturbative moduli spaces

TL;DR

This paper explores the connection between instanton corrections in supersymmetric gauge theories and integrable systems, revealing new relations involving the Thermodynamic Bethe Ansatz, Y-systems, and topological string functions.

Contribution

It identifies the contact potential with the free energy of an integrable system and links the Kahler potential to the Yang-Yang functional, advancing understanding of non-perturbative moduli spaces.

Findings

Contact potential equals the free energy of the integrable system.

Kahler potential corresponds to the Yang-Yang functional.

Y-system in the simplest case relates to the MacMahon function.

Abstract

Recently, an exact description of instanton corrections to the moduli spaces of 4d N=2 supersymmetric gauge theories compactified on a circle and Calabi-Yau compactifications of Type II superstring theories was found. The equations determining the instanton contributions turn out to have the form of Thermodynamic Bethe Ansatz. We explore further this relation and, in particular, we identify the contact potential of quaternionic string moduli space with the free energy of the integrable system and the Kahler potential of the gauge theory moduli space with the Yang-Yang functional. We also show that the corresponding S-matrix satisfies all usual constraints of 2d integrable models, including crossing and bootstrap, and derive the associated Y-system. Surprisingly, in the simplest case the Y-system is described by the MacMahon function relevant for crystal melting and topological strings.