Anomaly Inflow for Subsystem Symmetries

TL;DR

This paper explores 't Hooft anomalies and anomaly inflow in subsystem symmetries, including fracton models, demonstrating how anomalies are canceled by higher-dimensional SSPT phases and analyzing their boundary-bulk relations.

Contribution

It introduces a framework for understanding anomalies and anomaly inflow in subsystem symmetries, including classical continuum actions for bulk SSPT phases.

Findings

Anomalies for subsystem symmetries can be canceled by bulk SSPT phases.

Boundary anomalies are described by classical continuum actions.

Bulk SSPT phases depend on symmetry extension and foliation structure.

Abstract

We study 't Hooft anomalies and the related anomaly inflow for subsystem global symmetries. These symmetries and anomalies arise in a number of exotic systems, including models with fracton order such as the X-cube model. As is the case for ordinary global symmetries, anomalies for subsystem symmetries can be canceled by anomaly inflow from a bulk theory in one higher dimension; the corresponding bulk is therefore a non-trivial subsystem symmetry protected topological (SSPT) phase. We demonstrate these phenomena in several examples with continuous and discrete subsystem global symmetries, as well as time-reversal symmetry. For each example we describe the boundary anomaly, and present classical continuum actions for the corresponding bulk SSPT phases, which describe the response of background gauge fields associated with the subsystem symmetries. Interestingly, we show that the anomaly does not uniquely specify the bulk SSPT phase. In general, the latter may also depend on how the symmetry and the associated foliation structure on the boundary are extended into the bulk.