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

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 ${\mathrm{\Sigma }}_g$ 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.