Aspects of line operators of class S theories

TL;DR

This work develops a geometric framework for line operators in 4d $\mathcal{N}=2$ class ${\cal S}$ theories by mapping Wilson-'t Hooft lines to closed curves and webs on a Riemann surface. It introduces a Dirac pairing and mutual locality conditions, and uses closure of the OPE, along with maximality, to classify the allowed set of line operators, revealing how global gauge group form and discrete theta angles arise. Dualities are realized as mapping class group actions (S and T) on the homology coordinates, producing intricate duality webs across various genus and puncture configurations. The results connect line operator spectra to the geometry of the underlying surface, with implications for Hitchin moduli, integrable systems, and Nekrasov partition functions, and point to natural generalizations to non-full punctures and other gauge groups.

Abstract

Geometric picture of line operators of N=2 class S theories was found by imposing closure condition on operator product expansion (OPE) of line operators. In this paper, we first identify the geometric representation of ordinary Wilson-'t Hooft line operators of field theory, and study duality action on them. We further define a Dirac product between line operators and classify the allowed set of line operators by requiring: a: closure of OPE; b: mutual locality; c: maximality. Using above classifications, we find many distinct gauge theories associated with a single duality frame, and show explicitly that new possibilities correspond to the choice of global form of gauge group and discrete theta angles. We also study S and T duality actions relating those theories. In particular, we find very interesting duality webs for Maldacena-Nunez theory.

Figures (24)

• Figure 1: Duality action relating different gauge theory defined using genus two Riemann surface. Here $\textbf{1}$ means that gauge group is $SU(2)$, $2_0$ means the gauge group is $SU(2)/Z_2$, and $2_1$ means the gauge group is also $SU(2)/Z_2$, but there is a discrete $\theta$ angle associated with it. $S_i$ is the S duality transformation on gauge group and $T_i$ is the $T$ transformation.
• Figure 2: A surface operator of 6d $(2,0)$ theory is wrapped on a closed curve on Riemann surface and becomes line operator of 4d theory.
• Figure 3: Here we give some examples for line operator of $A_2$ theory on a fourth punctured sphere. Left: Wilson line with representation $R=2 w_1+2 w_2$ which is represented by colored closed curve. Right: A web operator formed by junctions.
• Figure 4: Skein relations which can be used to find OPE in a simple way. Here we only show some simple examples without showing the full set of skein relations.
• Figure 5: The use of skein relation in finding OPE of line operators of $A_2$ theory on a once punctured torus.