Wreathing, Discrete Gauging, and Non-invertible Symmetries

TL;DR

This paper explores the effects of discrete symmetries, anomalies, and non-invertible symmetries on the superconformal index of 3d $ ext{N}=4$ gauge theories, introducing a method to analyze symmetry gauging and wreathing operations.

Contribution

It provides a novel prescription for refining superconformal indices under discrete symmetry gauging, including wreathing, and applies it to complex quiver gauge theories.

Findings

Developed a method to refine indices with respect to Abelian discrete symmetries.

Analyzed the impact of 't Hooft anomalies and non-invertible symmetries on the index.

Validated the approach with examples involving 3d $ ext{N}=4$ gauge theories and wreathing operations.

Abstract

't Hooft anomalies of discrete global symmetries and gaugings thereof have rich mathematical structures and far-reaching physical consequences. We examine each subgroup $G$, up to automorphisms, of the permutation group $S_4$ that acts on the four legs of the affine $D_4$ quiver diagram, which is mirror dual to the 3d $\mathcal{N}=4$ $\mathrm{SU}(2)$ gauge theory with four flavours. These actions are studied in terms of how each permutation cycle acts on the superconformal index of the theory in question. We present a prescription for refining the index with respect to the fugacities associated with the Abelian discrete symmetries that are subgroups of $G$. This allows us to study sequential gauging of various subgroups of $G$ and construct symmetry webs. We study the effects of 't Hooft anomalies and non-invertible symmetries that arise from discrete gauging on the index. When the whole symmetry $G$ is gauged, our results are in perfect agreement with a type of discrete operations on the quiver, known as wreathing, discussed in the literature. We provide a general prescription for computing the index for any wreathed quivers that contain unitary or special unitary gauge groups. We demonstrate this in an example of the 3d $\mathcal{N}=4$ $\mathrm{U}(N)$ gauge theory with $n$ flavours and compare the results with gauging the charge conjugation symmetry associated with the flavour symmetry of such a theory.