Defect Networks and Supersymmetric Loop Operators

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.