Defect Networks and Supersymmetric Loop Operators

TL;DR

This paper explores the relationship between topological defect networks in Toda conformal field theory and supersymmetric loop operators in four-dimensional N=2 theories, revealing algebraic structures and connections to quantum groups.

Contribution

It introduces a framework linking defect networks in Toda CFT to supersymmetric loop operators, including the derivation of skein relations and their relation to quantum group representations.

Findings

Verlinde operators form an algebra governed by generalized skein relations.

Skein relations encode the representation theory of a quantum group.

The algebraic structures match with operator product expansions in supersymmetric theories.

Abstract

We consider topological defect networks with junctions in $A_{N-1}$ Toda CFT and the connection to supersymmetric loop operators in $\mathcal{N} = 2$ theories of class S on a four-sphere. Correlation functions in the presence of topological defect networks are computed by exploiting the monodromy of conformal blocks, generalising the notion of a Verlinde operator. Concentrating on a class of topological defects in $A_2$ Toda theory, we find that the Verlinde operators generate an algebra whose structure is determined by a set of generalised skein relations. These relations encode the representation theory of a quantum group. In the second half of the paper, we explore the dictionary between topological defect networks and supersymmetric loop operators in the $\mathcal{N}=2^*$ star theory by comparing to exact localisation computations. In this context, the the generalised skein relations are related to the operator product expansion of loop operators.