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

TL;DR

This work classifies the moduli spaces and generalized global symmetries of 3d ${\cal N}=5$ SCFTs, focusing on orthosymplectic ABJ theories and their discrete gaugings. It shows moduli spaces take the form $\mathbb{H}^{2N}/\Gamma$, where $\Gamma$ is a quaternionic reflection group or a $\mathbb{Z}_2$ extension, and provides a systematic method to obtain $\Gamma'$ when gauging a $\mathbb{Z}_2^{[0]}$ symmetry; Hilbert-series calculations agree with the corresponding limits of the superconformal index, and 't Hooft anomalies are encoded in both the index and the moduli space. The paper develops explicit symmetry webs for equal and unequal rank cases, including $D_8$ and $Q_8$-type structures, and extends the analysis to theories based on the $F(4)$ superalgebra. It also clarifies when discrete gaugings are anomalous and how these anomalies manifest in both the index and the moduli space, enriching the landscape of ${\cal N}=5$ SCFTs with Spin, O-, and Pin-type gauge groups. The results illuminate deep connections between anomalies, moduli-space quotient groups, and symmetry categories across a broad class of 3d supersymmetric theories.

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 $\mathbb{H}^{2N}/Γ$, where $Γ$ 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 $Γ$ 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 $Γ'$ 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 $Γ$. 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.

Figures (6)

• Figure 1: Lattice of subgroups of $D_8$, where each box represents a distinct conjugacy class of subgroups. Observe that the order two two-subnormal subgroups enjoy an inner automorphism generated by conjugations $x = s x s^{-1} = s x s$, $y = (r s) y (s r^3) = (r s) y (r s)$, where $x \in \langle rs \rangle$ and $y \in \langle s \rangle$.
• Figure 2: Lattice of subgroups of $Q_8$, where each box represents a distinct subgroup.
• Figure 3: The $D_8$ symmetry web for variants of the $\mathfrak{so}(2N)_{2k} \times \mathfrak{usp}(2N)_{-k}$ ABJ theory with $N$ even and $k$ even. Each arrow labelled by $\mathbb{Z}^{[0]}_{2,x}$ connecting two boxes denotes the gauging of the zero-form symmetry $\mathbb{Z}^{[0]}_{2,x}$. In each box, which is associated with a specific global form of the theory, we report the corresponding symmetry category and the quaternionic reflection group or its extension $\Gamma$ such that the moduli space is $\mathbb{H}^{2N}/\Gamma$. Note that the variant $[\mathrm{O}(2N)^-_{2k} \times \mathop{\mathrm{USp}}\nolimits(2N)_{-k}]/\mathbb{Z}_2$ is anomalous and not depicted here. We also emphasise that there are two distinct variants of the $\mathbb{Z}_2$ extension of the group $G_N(\widehat{D}_{k}, \mathbb{Z}_{2k})$ that are indicated by $\mathbb{Z}_2$ and $\mathbb{Z}_2'$. Moreover, in the special case of $N=2$, the group $\Gamma$ for $[\mathrm{Spin}(4)_{2k} \times \mathop{\mathrm{USp}}\nolimits(4)_{-k}]/\mathbb{Z}_2$, $\mathrm{Spin}(4)_{2k} \times \mathop{\mathrm{USp}}\nolimits(4)_{-k}$, and $\mathrm{Pin}(4)_{2k} \times \mathop{\mathrm{USp}}\nolimits(4)_{-k}$ turns out to be the quaternionic reflection groups $G_2(\widehat{D}_{2k},\mathbb{Z}_k)$, $G_2(\widehat{D}_{2k},\mathbb{Z}_{2k})$, and $G_2(\widehat{D}_{2k}, \widehat{D}_k)$ respectively; see \ref{['eq:specialcaseN2']}.
• Figure 4: The $D_8$ symmetry web for variants of the $\mathfrak{so}(2N)_{2k} \times \mathfrak{usp}(2N)_{-k}$ ABJ theory with $N$ even and $k$ odd. This diagram can be obtained from Figure \ref{['fig:D8Nevenkeven']} by exchanging ${\cal C}$ and ${\cal M} {\cal C}$ in the left part of the diagram. In each box, which is associated with a specific global form of the theory, we report the corresponding symmetry category and the quaternionic relection group or its extension $\Gamma$ such that the moduli space is $\mathbb{H}^{2N}/\Gamma$. Note that the variant $[\mathrm{O}(2N)^+_{2k} \times \mathop{\mathrm{USp}}\nolimits(2N)_{-k}]/\mathbb{Z}_2$ is anomalous and not depicted here. We emphasise that there are two distinct variants of the $\mathbb{Z}_2$ extension of the group $G_N(\widehat{D}_{k}, \mathbb{Z}_{2k})$ that are indicated by $\mathbb{Z}_2$ and $\mathbb{Z}_2'$. Moreover, in the special case of $N=2$, the group $\Gamma$ for $[\mathrm{Spin}(4)_{2k} \times \mathop{\mathrm{USp}}\nolimits(4)_{-k}]/\mathbb{Z}_2$, $\mathrm{Spin}(4)_{2k} \times \mathop{\mathrm{USp}}\nolimits(4)_{-k}$, and $\mathrm{Pin}(4)_{2k} \times \mathop{\mathrm{USp}}\nolimits(4)_{-k}$ turns out to be the quaternionic reflection groups $G_2(\widehat{D}_{2k},\mathbb{Z}_k)$, $G_2(\widehat{D}_{2k},\mathbb{Z}_{2k})$, and $G_2(\widehat{D}_{2k}, \widehat{D}_k)$ respectively; see \ref{['eq:specialcaseN2']}.
• Figure 5: The $D_8$ symmetry web for variants of the $\mathfrak{so}(2N)_{2k} \times \mathfrak{usp}(2N)_{-k}$ ABJ theory with $N$ odd and $k$ even. This diagram can be obtained from Figure \ref{['fig:D8Nevenkeven']} by exchanging ${\cal M}$ and ${\cal M} {\cal C}$ in the right part of the diagram. In each box, which is associated with a specific global form of the theory, we report the corresponding symmetry category and the quaternionic reflection group or its extension $\Gamma$ such that the moduli space is $\mathbb{H}^{2N}/\Gamma$. Note that the variant $[\mathrm{Spin}(2N)_{2k} \times \mathop{\mathrm{USp}}\nolimits(2N)_{-k}]/\mathbb{Z}_2$ is anomalous and not depicted here. We emphasise that there are two distinct variants of the $\mathbb{Z}_2$ extension of the group $G_N(\widehat{D}_{k}, \mathbb{Z}_{2k})$ that are indicated by $\mathbb{Z}_2$ and $\mathbb{Z}_2'$; for $N=2$ the latter is not a quaternionic reflection group and is explained around \ref{['gen:G2DkZ2k.Z2version2']}, whereas the former, associated with $\mathrm{Spin}(4)_{2k} \times \mathop{\mathrm{USp}}\nolimits(4)_{-k}$, is isomorphic to $G_2(\widehat{D}_{2k}, \mathbb{Z}_{2k})$. Moreover, for $\mathrm{Pin}(4)_{2k} \times \mathop{\mathrm{USp}}\nolimits(4)_{-k}$, the corresponding group turns out to be the quaternionic reflection group $G_2(\widehat{D}_{2k}, \widehat{D}_k)$; see \ref{['eq:specialcaseN2']}.