A unified view on symmetry, anomalous symmetry and non-invertible gravitational anomaly

TL;DR

This paper presents a unified framework for understanding symmetries, anomalous symmetries, and non-invertible gravitational anomalies in 1+1D models through multi-component partition functions and their transformations, revealing their role in constraining low energy dynamics.

Contribution

It introduces a systematic approach to treat symmetries and anomalies as non-invertible gravitational anomalies using partition functions and mapping class group transformations.

Findings

Symmetries and anomalies are described as non-invertible gravitational anomalies.

Partition functions encode low energy dynamics constrained by symmetries.

A unified view simplifies the classification of symmetry-related phenomena.

Abstract

In this paper, using 1+1D models as examples, we study symmetries and anomalous symmetries via multi-component partition functions obtained through symmetry twists, and their transformations under the mapping class group of spacetime. This point of view allows us to treat symmetries and anomalous symmetries as non-invertible gravitational anomalies (which are also described by multi-component partition functions, transforming covariantly under the mapping group transformations). This allows us to directly see how symmetry and anomalous symmetry constraint the low energy dynamics of the systems, since the low energy dynamics is directly encoded in the partition functions. More generally, symmetries, anomalous symmetries, non-invertible gravitational anomalies, and their combinations, can all be viewed as constraints on low energy dynamics. In this paper, we demonstrate that they all can be viewed uniformally and systematically as pure (non-invertible) gravitational anomalies.