Splitting of surface defect partition functions and integrable systems

TL;DR

This paper explores how surface defect partition functions in a specific supersymmetric gauge theory split into parts at special Coulomb moduli, revealing differential equations and a Bethe/gauge correspondence for each part.

Contribution

It demonstrates the splitting of surface defect partition functions at special Coulomb moduli and establishes the Bethe/gauge correspondence for each component.

Findings

Partition functions split into multiple parts at special Coulomb moduli.

Differential equations of Schrödinger and Baxter types are satisfied by these parts.

Bethe/gauge dictionary is recovered for each summand.

Abstract

We study Bethe/gauge correspondence at the special locus of Coulomb moduli where the integrable system exhibits the splitting of degenerate levels. For this investigation, we consider the four-dimensional pure $\mathcal{N}=2$ supersymmetric $U(N)$ gauge theory, with a half-BPS surface defect constructed with the help of an orbifold or a degenerate gauge vertex. We show that the non-perturbative Dyson-Schwinger equations imply the Schr\"odinger-type and the Baxter-type differential equations satisfied by the respective surface defect partition functions. At the special locus of Coulomb moduli the surface defect partition function splits into parts. We recover the Bethe/gauge dictionary for each summand.