Defect Networks and Supersymmetric Loop Operators

TL;DR

The paper establishes a detailed correspondence between topological defect networks in A_{N-1} Toda CFT and supersymmetric loop operators in class S N=2 theories on S^4, by formulating Verlinde-type operators for networks and showing they obey generalised skein relations tied to quantum groups U_{\\mathfrak{q}}(\\mathfrak{sl}(N)). Focusing on A_2, it derives explicit network-induced operators for Wilson, ’t Hooft, and dyonic loops, and demonstrates their operator product expansions via generalized skein relations, matching localization results for N=2^*. The work thus provides a concrete dictionary between four-dimensional loop observables and two-dimensional defect networks, including detailed OPE and commutation structures, and casts loop algebras as spiders for quantum groups. These results illuminate the algebraic structure behind dualities and offer a robust framework to compute loop operator products in higher-rank theories using Toda/CFT data.

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.

Figures (13)

• Figure 1: An example of a defect network in $A_2$ Toda theory on a torus with simple puncture. It corresponds to the dyonic loop operator labelled by the weights $(-\omega_2,\omega_1)$ in the $\mathcal{N}=2^*$ theory.
• Figure 2: The ingredients for topological defect networks in Toda theory of type $A_2$ are a) oriented lines and b) incoming and outgoing vertices.
• Figure 3: A topological defect a) labelled by the representation with momentum $\mu=-b\omega_j$ wrapping the tube of a pants decomposition with momentum $\alpha$, and the sequence b) of fusion and braiding operations needed to compute the Verlinde operator.
• Figure 4: The four-point correlation function of with two non-degenerate momentum $\alpha_1$ and $\bar{\alpha}_2 = 2Q-\alpha_2$, one semi-degenerate momentum of the form $\nu\equiv-\kappa h_N$ where $\kappa\in \mathbb{R}$, and one completely degenerate momentum $\mu\equiv-bh_1$. In what follows we omit the arrows on conformal block diagrams, with the understanding that they are pointing downwards and to the right, as shown here.
• Figure 5: In $A_2$ Toda theory, the four-point correlator with two primary fields of equal degenerate momentum $\mu=-b\omega_1$ is related to that standard four-point function with the specialisation $\kappa = 3q+b$ by applying the Weyl transformation $w:h_j \to h_{j+1}$. The conformal blocks of the above four-point correlators are therefore equal.