Darboux coordinates, Yang-Yang functional, and gauge theory

TL;DR

This paper links the geometry of moduli spaces of flat connections and opers with supersymmetric gauge theories, providing a geometric interpretation of Yang-Yang functionals and connecting them to instanton counting.

Contribution

It introduces a geometric framework for Yang-Yang functionals in quantum Hitchin systems using Darboux coordinates and moduli space geometry, relating gauge theory and integrable systems.

Findings

Identifies the generating function of SL(2)-opers with the superpotential of 4D N=2 theories.

Defines Yang-Yang functionals via classical geometry of moduli spaces.

Connects gauge theory instanton counting with the geometry of local systems.

Abstract

The moduli space of SL(2) flat connections on a punctured Riemann surface with the fixed conjugacy classes of the monodromies around the punctures is endowed with a system of holomorphic Darboux coordinates, in which the generating function of the variety of SL(2)-opers is identified with the universal part of the effective twisted superpotential of the corresponding four dimensional N=2 supersymmetric theory subject to the two-dimensional Omega-deformation. This allows to give a definition of the Yang-Yang functionals for the quantum Hitchin system in terms of the classical geometry of the moduli space of local systems for the dual gauge group, and connect it to the instanton counting of the four dimensional gauge theories, in the rank one case.