"Zoology" of non-invertible duality defects: the view from class $\mathcal{S}$

TL;DR

The paper investigates non-invertible duality defects in 4d $\mathcal{N}=2$ class $\mathcal{S}$ theories arising from 6d $\mathcal{N}=(2,0)$ on $\Sigma_g$, generalizing known duality defects from $\mathcal{N}=4$ SYM to theories with larger one-form symmetry. It develops two complementary frameworks to determine the fusion algebra: (i) a 4d approach based on discrete topological manipulations governed by the central extension $\mathrm{Sp}(2g,\mathbb{Z}_N)_T$, and (ii) a 5d symmetry TFT that encodes bulk topological data and boundary conditions; both culminate in a group-like fusion structure with additional data from decoupled TQFTs $\mathcal{N}^{r,\pm}$ and condensation defects $\mathcal{C}^{\mathcal{A}}$. A central technical novelty is the introduction of the rank of a non-invertible duality defect, which almost fixes the fusion rules by constraining the surviving line operators via $\mathcal{K}=\mathcal{L}\wedge M^{-1}\mathcal{L}$ and its kernel structures. The paper also provides explicit applications for genus $g=2$ with cyclic $\mathbb{Z}_{4g+2}$ and non-abelian dihedral $D_{4g+4}$ automorphism groups, revealing nontrivial, sometimes noncommutative fusion patterns and validating the 5d/6d holographic perspective against the 4d construction. Overall, the work offers a coherent program to classify and compute non-invertible duality defects in class $\mathcal{S}$ theories and connects discrete gauging, 2-form symmetries, and symmetry TFT data into a unified picture with concrete holographic checks.

Abstract

We study generalizations of the non-invertible duality defects present in $\mathcal{N} = 4$ SU(N) SYM by studying theories with larger duality groups. We focus on 4d $\mathcal{N} = 2$ theories of class $\mathcal{S}$ obtained by the dimensional reduction of the 6d $\mathcal{N} = (2, 0)$ theory of $A_{N-1}$ type on a Riemann surface $Σ_g$ without punctures. We discuss their non-invertible duality symmetries and provide two ways to compute their fusion algebra: either using discrete topological manipulations or a 5d TQFT description. We also introduce the concept of "rank" of a non-invertible duality symmetry and show how it can be used to (almost) completely fix the fusion algebra with little computational effort.