Anomalies and symmetry fractionalization

Abstract

We study ordinary, zero-form symmetry GG and its anomalies in a system with a one-form symmetry \GammaΓ. In a theory with one-form symmetry, the action of GG on charged line operators is not completely determined, and additional data, a fractionalization class, needs to be specified. Distinct choices of a fractionalization class can result in different values for the anomalies of GG if the theory has an anomaly involving \GammaΓ. Therefore, the computation of the ’t Hooft anomaly for an ordinary symmetry GG generally requires first discovering the one-form symmetry \GammaΓ of the physical system. We show that the multiple values of the anomaly for GG can be realized by twisted gauge transformations, since twisted gauge transformations shift fractionalization classes. We illustrate these ideas in QCD theories in diverse dimensions. We successfully match the anomalies of time-reversal symmetries in 2+1d2+1d gauge theories, across the different fractionalization classes, with previous conjectures for the infrared phases of such strongly coupled theories, and also provide new checks of these proposals. We perform consistency checks of recent proposals about two-dimensional adjoint QCD and present new results about the anomaly of the axial \mathbb{Z}_{2N}ℤ2N symmetry in 3+1d3+1d{\cal N}=1 super-Yang-Mills. Finally, we study fractionalization classes that lead to 2-group symmetry, both in QCD-like theories, and in 2+1d2+1d\mathbb{Z}_2ℤ2 gauge theory.