Symmetries and Anomalies of (1+1)d Theories: 2-groups and Symmetry Fractionalization

TL;DR

This paper explores the complex interactions of discrete symmetries in (1+1)d theories, focusing on 2-group symmetries, symmetry fractionalization, and their anomalies, with implications for topological manipulations and theory decomposition.

Contribution

It provides a detailed analysis of 2-group symmetries and symmetry fractionalization in (1+1)d, including anomaly calculations and effects of gauging one-form symmetries.

Findings

Identification of multiple sectors in (1+1)d theories due to symmetry interactions.

Calculation of anomalies using spectral sequences for 2-group and fractionalized symmetries.

Insights into topological operator manipulations and their (1+1)d effects via coupling to (2+1)d theories.

Abstract

We investigate the interactions of discrete zero-form and one-form global symmetries in (1+1)d theories. Focus is put on the interactions that the symmetries can have on each other, which in this low dimension result in 2-group symmetries or symmetry fractionalization. A large part of the discussion will be to understand a major feature in (1+1)d: the multiple sectors into which a theory decomposes. We perform gauging of the one-form symmetry, and remark on the effects this has on our theories, especially in the case when there is a global 2-group symmetry. We also implement the spectral sequence to calculate anomalies for the 2-group theories and symmetry fractionalized theory in the bosonic and fermionic cases. Lastly, we discuss topological manipulations on the operators which implement the symmetries, and draw insights on the (1+1)d effects of such manipulations by coupling to a bulk (2+1)d theory.