Holomorphic reduction of N=2 gauge theories, Wilson-'t Hooft operators, and S-duality

TL;DR

This paper explores a twisted N=2 superconformal gauge theory on a product of Riemann surfaces, revealing its topological and holomorphic properties, its relation to the geometric Langlands duality, and the behavior of Wilson-'t Hooft operators.

Contribution

It establishes a connection between twisted N=2 gauge theories and the geometric Langlands duality through the study of Wilson-'t Hooft operators and S-duality.

Findings

The theory is topological along C and holomorphic along Sigma.

Correlators of loop operators depend holomorphically and are gauge coupling independent.

Duality acts as autoequivalences on the derived category of the moduli space.

Abstract

We study twisted N=2 superconformal gauge theory on a product of two Riemann surfaces Sigma and C. The twisted theory is topological along C and holomorphic along Sigma and does not depend on the gauge coupling or theta-angle. Upon Kaluza-Klein reduction along Sigma, it becomes equivalent to a topological B-model on C whose target is the moduli space MV of nonabelian vortex equations on Sigma. The N=2 S-duality conjecture implies that the duality group acts by autoequivalences on the derived category of MV. This statement can be regarded as an N=2 counterpart of the geometric Langlands duality. We show that the twisted theory admits Wilson-'t Hooft loop operators labelled by both electric and magnetic weights. Correlators of these loop operators depend holomorphically on coordinates and are independent of the gauge coupling. Thus the twisted theory provides a convenient framework for studying the Operator Product Expansion of general Wilson-'t Hooft loop operators.