Line operators from M-branes on compact Riemann surfaces

TL;DR

This paper determines the charge lattice of line operators in class S theories derived from M5-branes on Riemann surfaces, using geometric and quantum constraints to ensure mutual locality and gauge group structure.

Contribution

It provides a geometric and quantum framework to identify the charge lattice of line operators in class S theories, clarifying their mutual locality and gauge group structure.

Findings

Charge lattice of line operators explicitly determined.

Mutual locality conditions imposed by quantum constraints.

Structure of the four-dimensional gauge group clarified.

Abstract

In this paper, we determine the charge lattice of mutually local Wilson and 't Hooft line operators for class S theories living on M5-branes wrapped on compact Riemann surfaces. The main ingredients of our analysis are the fundamental group of the N-cover of the Riemann surface, and a quantum constraint on the six-dimensional theory. This latter plays a central role in excluding some of the possible lattices and imposing consistency conditions on the charges. This construction gives a geometric explanation for the mutual locality among the lines, fixing their charge lattice and the structure of the four-dimensional gauge group.