Higher Symmetries of 5d Orbifold SCFTs

TL;DR

This work develops a geometry-based framework to determine higher-form symmetries of 5d SCFTs arising from M-theory on orbifolds $\mathbb{C}^3/\Gamma$ with $\Gamma\subset SU(3)$. It presents two complementary, resolution-independent methods: (i) an algebraic-topology approach that computes the defect group from $\pi_1(S^5/\Gamma)$ via $\pi_1(S^5/\Gamma) \cong \Gamma/H_{\Gamma,f_\Gamma}$ and $\mathbb{D}^{(1)} = \text{Ab}[\pi_1(S^5/\Gamma)]$, and (ii) a quiver approach that uses the 5d BPS quiver to extract the 1-form/2-form data from the Dirac pairing matrix $B$ through $\text{Tor}(\text{coker}(B))$, yielding identical results. The paper applies these methods to abelian, U(2)-derived, and larger transitive subgroups, revealing a spectrum of higher-form symmetries (including cases with $\mathbb{Z}_5$, $\mathbb{Z}_9$, and trivial groups) and highlighting 2-group structures linked to the abelianization $\mathrm{Ab}[\Gamma]$. It demonstrates that the symmetry data is intrinsic to the SCFT fixed point and outlines avenues to compute the Postnikov class from geometry and to extend the analysis to lower-dimensional compactifications and non-supersymmetric orbifolds.

Abstract

We determine the higher symmetries of 5d SCFTs engineered from M-theory on a $\mathbb{C}^3 / Γ$ background for $Γ$ a finite subgroup of $SU(3)$. This resolves a longstanding question as to how to extract this data when the resulting singularity is non-toric (when $Γ$ is non-abelian) and/or not isolated (when the action of $Γ$ has fixed loci). The BPS states of the theory are encoded in a 1d quiver quantum mechanics gauge theory which determines the possible 1-form and 2-form symmetries. We also show that this same data can also be extracted by a direct computation of the corresponding defect group associated with the orbifold singularity. Both methods agree, and these computations do not rely on the existence of a resolution of the singularity. We also observe that when the geometry faithfully captures the global 0-form symmetry, the abelianization of $Γ$ detects a 2-group structure (when present). As such, this establishes that all of this data is indeed intrinsic to the superconformal fixed point rather than being an emergent property of an IR gauge theory phase.

Figures (5)

• Figure 1: Left: the orbifold quiver for the $\mathbb{C}^3/\mathbb{Z}_6$ theory generated by $g = \frac{1}{6}(1,2,3)$. Right: The orbifold quiver for the $\mathbb{C}^3/\mathbb{Z}_{12}$ theory generated by $g = \frac{1}{12}(1,2,9)$.
• Figure 2: The orbifold quiver for the $\mathbb{C}^3/D_{5,3}$ theory.
• Figure 3: The quiver for the $\mathbb{C}^3/T_3$ orbifold theory.
• Figure 4: Left: The BPS quiver corresponding to the orbifold group $T_5$; Right: The BPS quiver corresponding to the orbifold group $T_7$. Notice that we can recognize a box product-like structure in the quiver with an affine $\hat{E}_6$ structure.
• Figure 5: Left: The BPS quiver corresponding to the orbifold group $O_5$; Right: The BPS quiver corresponding to the orbifold group $O_7$. Notice that we can recognize a box product-like structure in the quiver with an affine $\hat{E}_7$ structure.