Higher Symmetries of 5d Orbifold SCFTs

Abstract

We determine the higher symmetries of 5d SCFTs engineered from M-theory on a 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.

0-Form, 1-Form and 2-Group Symmetries via Cutting and Gluing of Orbifolds

“…the (possibly singular) boundary geometry S 5 /Γ. It was also conjectured in [10] that the structure of candidate 2-group symmetries is closely correlated with the abelianization of Γ itself.…”

“…This was used in [7][8][9] to explicitly construct the resulting (relative) homology cycles, though it rapidly becomes quite complicated to explicitly track all of this data. In [10] an alternative approach was developed for 5D SCFTs obtained from the orbifold singularities C 3 /Γ for Γ a finite subgroup of SU (3). Instead of explicitly performing any resolutions, the data of 1-form symmetries was extracted from the adjacency matrix of the corresponding 5D BPS quiver (see [11]), as well as from the fundamental group for Figure 1: Depiction of an SQFT realized at a localized region of a non-compact geometry X, with boundary ∂X.…”

Snowmass White Paper on SCFTs

“…As in the case of 6D SCFTs, there need not exist a gauge theory phase in the moduli space, and the SCFTs can be obtained by taking singular limits of brane and geometric constructions. See [56,62,[75][76][77][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92][93][94] for additional constructions of 5D SCFTs.…”

“…If this were true, the question would reduce to determining all possible moduli space flows in that more limited setting. Examples which resist an obvious embedding in an elliptically fibered Calabi-Yau threefold include orbifolds of the form C 3 /Γ for Γ a finite subgroup of SU (3) [84,93,94,158]. That said, these examples may be connected to other theories via a flow in moduli space.…”

“…There can be continuous higher symmetry in non-conformal supersymmetric theories, and its presence is an obstruction to conformal invariance anywhere along the RG flow unless the symmetry is accidental in the IR and explicitly broken in the UV [125,430]. SCFTs can have interesting discrete higher-form symmetries, and this has been an active area of recent research: recent work in six dimensions includes [159,386,[431][432][433][434][435]; in five dimensions includes [94,392,434,436,437]; in four dimensions includes [393,[438][439][440][441]; and in three dimensions includes [442][443][444][445][446][447][448].…”