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

Abstract

Orbifold singularities of M-theory constitute the building blocks of a broad class of supersymmetric quantum field theories (SQFTs). In this paper we show how the local data of these geometries determines global data on the resulting higher symmetries of these systems. In particular, via a process of cutting and gluing, we show how local orbifold singularities encode the 0-form, 1-form and 2-group symmetries of the resulting SQFTs. Geometrically, this is obtained from the possible singularities which extend to the boundary of the non-compact geometry. The resulting category of boundary conditions then captures these symmetries, and is equivalently specified by the orbifold homology of the boundary geometry. We illustrate these general points in the context of a number of examples, including 5D superconformal field theories engineered via orbifold singularities, 5D gauge theories engineered via singular elliptically fibered Calabi-Yau threefolds, as well as 4D SQCD-like theories engineered via M-theory on non-compact G 2 spaces.

6d SCFTs, Center-Flavor Symmetries, and Stiefel--Whitney Compactifications

“…Since for this generator, all −c 2 (F i ) ≡ N −1 2N w 2 fractionalize equally, they cancel out for each tensor t i . Therefore, the non-Abelian symmetry group is [SU (N ) L × i SU (N ) (i) × SU (N ) R ]/Z N , and the non-Abelian flavor symmetry of the SCFT is [SU (N ) L ×SU (N ) R ]/Z N , which agrees with known results [36,82]. As an additional comment, we note that this case also has an overall u(1) flavor symmetry [36,64], so we will revisit it when we discuss Abelian symmetry factors.…”

“…As before, we leave the spacetime symmetries implicit. The answer is naive, in the sense that this analysis does not distinguish between symmetries acting on genuine local operators, and those which are only defined as the endpoints of line operators (see [80,82,105,106]). Indeed, on general grounds, we expect that the actual zero-form symmetry group is quotiented by a subgroup of the common center for these factors.…”

“…We now turn to examples illustrating how we extract the non-Abelian flavor symmetries. Let us also note that recently in [82], geometric methods were developed to directly extract the global symmetry group for 5d conformal matter, i.e., the circle reduction of 6d conformal matter. Our bottom up analysis agrees with the results found there.…”

“…This quotient can also act on the spacetime symmetries since the supercharges transform as spacetime spinors and R-symmetry spinors. The combined action on the gauge and flavor symmetry is often referred to as a "center-gauge-flavor symmetry", generalizing the notion of "center-flavor symmetry" [65][66][67][68][69][70][71][72][73][74][75][76][77][78][79][80][81][82]. Of course, from the perspective of the 6d SCFT, the defining data only makes reference to the global symmetries, and the same quotient, suitably projected, realizes the continuous part of the global symmetry group…”

“…This also amounts (upon projecting onto the global symmetry factors) to a prediction for the global continuous zero-form group of the 6d SCFT. Extracting this data directly from the corresponding F-/M-theory background geometry [82] was recently carried out for a number of 5d supersymmetric quantum field theories obtained from circle reduction of the tensor branch of a 6d SCFT, and the "bottom up" approach developed here agrees with the "top down" results obtained in [82]. From a bottom up perspective, we simply work on on the tensor branch where we have access to the large symmetry transformations of the system, and the correlated response from transformations on the chiral two-forms of the effective field theory.…”