Loop operators and S-duality from curves on Riemann surfaces

TL;DR

This paper classifies Wilson-'t Hooft loop operators in certain N=2 superconformal theories using curves on Riemann surfaces, providing explicit predictions for S-duality actions and matching homotopy classifications.

Contribution

It establishes a correspondence between loop operators' charges and homotopy classes of curves on Riemann surfaces in Gaiotto theories, offering a geometric understanding of S-duality.

Findings

Classification of loop operators via homotopy classes

Explicit predictions for S-duality transformations

Matching of charges with geometric curve classes

Abstract

We study Wilson-'t Hooft loop operators in a class of N=2 superconformal field theories recently introduced by Gaiotto. In the case that the gauge group is a product of SU(2) groups, we classify all possible loop operators in terms of their electric and magnetic charges subject to the Dirac quantization condition. We then show that this precisely matches Dehn's classification of homotopy classes of non-self-intersecting curves on an associated Riemann surface--the same surface which characterizes the gauge theory. Our analysis provides an explicit prediction for the action of S-duality on loop operators in these theories which we check against the known duality transformation in several examples.