Interplay of Generalised Symmetries and Moduli Spaces in 3d $\mathcal{N}=5$ SCFTs

TL;DR

This paper explores the structure of moduli spaces and generalized symmetries in 3d $ ext{N}=5$ superconformal field theories, extending classifications, constructing new symmetry groups, and verifying results through Hilbert series and superconformal indices.

Contribution

It introduces a systematic method to determine the moduli space symmetry group after gauging $ ext{Z}_2$ symmetries and extends the classification of $ ext{N}=5$ moduli spaces to theories with various gauge groups.

Findings

Moduli spaces are governed by a $ ext{Z}_2$ central extension of quaternionic reflection groups.

Constructed explicit symmetry webs for $ ext{so}(2N)_{2k} imes ext{usp}(2N)_{-k}$ theories with different parities.

Verified the Hilbert series against superconformal index limits, confirming the theoretical predictions.

Abstract

The moduli space and generalised global symmetries of 3d $\mathcal{N} = 5$ superconformal field theories are investigated, with a focus on the orthosymplectic ABJ theories and their discrete gauging variants. We extend the known classification of $\mathcal{N}=5$ moduli spaces as orbifolds $\mathbb{H}^{2N}/\Gamma$, where $\Gamma$ is a quaternionic reflection group, to theories incorporating $\mathrm{Spin}$, $\mathrm{O}^-$, and $\mathrm{Pin}$-type gauge groups. In these cases, we find that the moduli space is governed not by $\Gamma$ itself, but by a $\mathbb{Z}_2$ central extension thereof, for which we explicitly describe the generators. We provide a systematic method to construct the group $\Gamma'$ governing the moduli space of a theory $\mathcal{T}'$ obtained by gauging a $\mathbb{Z}_2$ zero-form symmetry of an original theory $\mathcal{T}$. This is achieved by identifying the specific generator that must be added to $\Gamma$. We compute the Hilbert series for these moduli spaces and verify them against the corresponding limits of the superconformal index, finding perfect agreement. We also discuss how 't Hooft anomalies for the zero-form symmetries manifest in the superconformal index and the moduli space. Furthermore, we revisit the symmetry category of the $\mathfrak{so}(2N)_{2k} \times \mathfrak{usp}(2N)_{-k}$ theories. Building on previous work that identified the symmetry category for all parities of $N$ and $k$, we provide the explicit symmetry webs for the opposite parity $D_8$ case. We find that the details of these webs differ from the previously studied $D_8$ webs corresponding to the both even parity case. Finally, we analyse theories with unequal ranks, those containing the $\mathfrak{so}(2N+1)$ gauge algebra, and the two SCFT variants based on the $F(4)$ superalgebra.