Anomalies and Symmetry Fractionalization

TL;DR

This paper explores how one-form symmetries and their fractionalization affect anomalies in gauge theories, providing new insights into anomaly matching, fractionalization classes, and their implications in various dimensions.

Contribution

It introduces the concept of fractionalization classes for one-form symmetries and demonstrates their impact on anomalies, with applications to QCD and super-Yang-Mills theories.

Findings

Matched anomalies of time-reversal symmetries in 2+1d gauge theories across fractionalization classes

Identified fractionalization classes leading to 2-group symmetry in QCD-like theories

Provided new checks and results on anomalies in 3+1d super-Yang-Mills

Abstract

We study ordinary, zero-form symmetry $G$ and its anomalies in a system with a one-form symmetry $\Gamma$. In a theory with one-form symmetry, the action of $G$ 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 $G$ if the theory has an anomaly involving $\Gamma$. Therefore, the computation of the 't Hooft anomaly for an ordinary symmetry $G$ generally requires first discovering the one-form symmetry $\Gamma$ of the physical system. We show that the multiple values of the anomaly for $G$ 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 $2+1d$ 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 $\mathbb{Z}_{2N}$ symmetry in $3+1d$ ${\cal N}=1$ super-Yang-Mills. Finally, we study fractionalization classes that lead to 2-group symmetry, both in QCD-like theories, and in $2+1d$ $\mathbb{Z}_2$ gauge theory.