We study the defect groups of $$ {D}_p^b $$ D p b (G) theories using geometric engineering and BPS quivers. In the simple case when b = h∨(G), we use the BPS quivers of the theory to see that the defect group is compatible with a known Maruyoshi-Song flow. To extend to the case where b ≠ h∨(G), we use a similar Maruyoshi-Song flow to conjecture that the defect groups of $$ {D}_p^b $$ D p b (G) theories are given by those of G(b)[k] theories. In the cases of G = An, E6, E8 we cross check our result by calculating the BPS quivers of the G(b)[k] theories and looking at the cokernel of their intersection matrix.

We determine the higher symmetries of 5D SCFTs engineered from M-theory on a C 3 =Γ background for Γ a finite subgroup of SUð3Þ. This resolves a longstanding question as to how to extract this data when the resulting singularity is nontoric (when Γ is non-Abelian) and/or not isolated (when the action of Γ has fixed loci). The Bogomol'nyi-Prasad-Sommerfield (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.

When a quantum field theory in d-spacetime dimensions possesses a global $$(d-1)$$ ( d - 1 ) -form symmetry, it can decompose into disjoint unions of other theories. This is reflected in the physical quantities of the theory and can be used to study properties of the constituent theories. In this note we highlight the equivalence between the decomposition of orbifold $$\sigma $$ σ -models and disconnected McKay quivers. Specifically, we show in numerous examples that each component of a McKay quiver can be given definitive geometric meaning through the decomposition formulae. In addition, we give a purely group and representation theoretic derivation of the quivers for the cases where the trivially acting part of the orbifold group is central. As expected, the resulting quivers are compatible with the case of $$\sigma $$ σ -models on ‘banded’ gerbes.

When a quantum field theory in d-spacetime dimensions possesses a global (d − 1)-form symmetry, it can decompose into disjoint unions of other theories. This is reflected in the physical quantities of the theory and can be used to study properties of the constituent theories. In this note we highlight the equivalence between the decomposition of orbifold σmodels and disconnected McKay quivers. Specifically, we show in numerous examples that each component of a McKay quiver can be given definitive geometric meaning through the decomposition formulae. In addition, we give a purely group and representation theoretic derivation of the quivers for the cases where the trivially acting part of the orbifold group is central. As expected, the resulting quivers are compatible with the case of σ -models on 'banded' gerbes.

In this paper we introduce the characteristic dimension of a four dimensional $${{\mathcal {N}}}=2$$ N = 2 superconformal field theory, which is an extraordinary simple invariant determined by the scaling dimensions of its Coulomb branch operators. We prove that only nine values of the characteristic dimension are allowed, $$-\infty $$ - ∞ , 1 ,6/5, 4/3, 3/2, 2, 3, 4, and 6, thus giving a new organizing principle to the vast landscape of 4d $${\mathcal {N}}=2$$ N = 2 SCFTs. Whenever the characteristic dimension differs from 1 or 2, only very constrained special Kähler geometries (i.e. isotrivial, diagonal and rigid) are compatible with the corresponding set of Coulomb branch dimensions and extremely special, maximally strongly coupled, BPS spectra are allowed for the theories which realize them. Our discussion applies to superconformal field theories of arbitrary rank, i.e. with Coulomb branches of any complex dimension. Along the way, we predict the existence of new $${{\mathcal {N}}}=3$$ N = 3 theories of rank two with non-trivial one-form symmetries.