Opers, surface defects, and Yang-Yang functional

TL;DR

This paper investigates the non-perturbative equations governing partition functions of supersymmetric gauge theories with surface defects, introduces generalized Darboux coordinates, and proves a conjecture relating opers to superpotentials.

Contribution

It introduces generalized Darboux coordinates for flat connections and proves the NRS conjecture for $SL(3)$, extending previous results from the $SL(2)$ case.

Findings

Relation between different defect partition functions via analytic continuation

Introduction of Darboux coordinates on moduli space of flat connections

Proof of the NRS conjecture for $SL(3)$ case

Abstract

We explore the non-perturbative Dyson-Schwinger equations obeyed by the partition functions of the $\Omega$-deformed $\mathcal{N}=2, d=4$ supersymmetric linear quiver gauge theories in the presence of surface defects. We demonstrate that the partition functions of different types of defects (orbifold or vortex strings) are related by analytic continuation. We introduce Darboux coordinates on a patch of the moduli space of flat $SL(N)$-connections on a sphere with special punctures, which generalize the NRS coordinates defined in the $SL(2)$ case. Finally, we compare the generating function of the Lagrangian variety of opers in these Darboux coordinates with the effective twisted superpotential of the linear quiver theory in the two-dimensional $\Omega$-background, thereby proving the NRS conjecture and its generalization to the $SL(3)$ case.