Bethe/Gauge correspondence in odd dimension: modular double, non-perturbative corrections and open topological strings

TL;DR

This paper extends the Bethe/Gauge correspondence to five and three dimensions by incorporating non-perturbative effects and modular double structures, connecting gauge theories on compact spaces with topological string theory and integrable systems.

Contribution

It develops a full Bethe/Gauge correspondence dictionary on $S^5$ and $S^3$, incorporating non-perturbative corrections and modular double structures, especially for relativistic Toda chains.

Findings

Established Bethe/Gauge correspondence on $S^5$ and $S^3$.

Connected eigenfunctions to non-perturbatively completed topological strings.

Highlighted the role of Faddeev's modular double in odd-dimensional gauge theories.

Abstract

Bethe/Gauge correspondence as it is usually stated is ill-defined in five dimensions and needs a "non-perturbative" completion; a related problem also appears in three dimensions. It has been suggested that this problem, probably due to incompleteness of Omega background regularization in odd dimension, may be solved if we consider gauge theory on compact $S^5$ and $S^3$ geometries. We will develop this idea further by giving a full Bethe/Gauge correspondence dictionary on $S^5$ and $S^3$ focussing mainly on the eigenfunctions of (open and closed) relativistic 2-particle Toda chain and its quantized spectral curve: these are most properly written in terms of non-perturbatively completed NS open topological strings. A key ingredient is Faddeev's modular double structure which is naturally implemented by the $S^5$ and $S^3$ geometries.