Supersymmetric gauge theories, quantization of moduli spaces of flat connections, and conformal field theory

TL;DR

This paper explores the deep connection between supersymmetric gauge theories and conformal field theory by quantizing moduli spaces of flat connections, revealing new insights into instanton partition functions and Liouville conformal blocks.

Contribution

It provides a derivation of the gauge theory and conformal field theory correspondence using the quantization of moduli spaces, linking Riemann-Hilbert problems with Liouville theory.

Findings

Quantum moduli spaces serve as a non-perturbative framework for gauge theories.

Instanton partition functions are solutions to a Riemann-Hilbert problem.

The relation kernel is identified with Liouville conformal blocks.

Abstract

We will propose a derivation of the correspondence between certain gauge theories with N=2 supersymmetry and conformal field theory discovered by Alday, Gaiotto and Tachikawa in the spirit of Seiberg-Witten theory. Based on certain results from the literature we argue that the quantum theory of the moduli spaces of flat SL(2,R)-connections represents a non-perturbative "skeleton" of the gauge theory, protected by supersymmetry. It follows that instanton partition functions can be characterized as solutions to a Riemann-Hilbert type problem. In order to solve it, we describe the quantization of the moduli spaces of flat connections explicitly in terms of two natural sets of Darboux coordinates. The kernel describing the relation between the two pictures represents the solution to the Riemann Hilbert problem, and is naturally identified with the Liouville conformal blocks.