Higher SPT's and a generalization of anomaly in-flow

TL;DR

This work analyzes 't Hooft anomalies for finite symmetries and shows that not all anomalies admit traditional anomaly-in-flow to a conventional SPT bulk; some are too severe and require higher-symmetry (d-group/2-group) protections, i.e., higher SPTs, to realize a consistent boundary theory.Using a concrete 2+1d Dijkgraaf-Witten model with $G_{gauge}={\mathbb Z}/3$ and $G_{global}={\mathbb Z}/3\times{\mathbb Z}/3$, the authors demonstrate that upgrading to a 2-group symmetry with a 1-form gauge field permits a local 4d bulk action that cancels the boundary anomaly via anomaly in-flow, through a Serre spectral sequence analysis.They construct lattice realizations via a decorated Walker-Wang bulk that encodes the 2-SPT action, and they elucidate how nonlocal boundary actions can be tamed by higher symmetry, enabling finite-depth quantum circuits to implement restricted symmetry actions on regions with boundary.Beyond the 2-SPT case, the paper generalizes to Dijkgraaf-Witten theories in arbitrary dimensions and to general 2+1d TQFTs, showing that anomalous gaugings can be understood as obstructions that are resolved by higher-group inflows or, when necessary, by embedding into 2-SPT or higher SPT boundaries.

Abstract

Symmetry protected topological (SPT) phases of bosons in $d$ spatial dimensions have been characterized by the action of the protecting global symmetry $G$ on their boundary. The symmetry acts on the boundary in a way that would be impossible to realize in a purely $d-1$ dimensional system i.e., without the bulk. This is often formalized by saying the $G$ symmetry is anomalous when the boundary theory is gauged. Simultaneously gauging the symmetry on the boundary and in the bulk yields a gauge-invariant composite system. One says there is an anomaly in-flow from the boundary to the bulk. Recently it has been appreciated that some anomalies are too severe to be regulated by an SPT bulk. These do not satisfy anomaly in-flow in the traditional sense. However, we show that these anomalous systems can be regulated as the symmetric boundaries of the newly discovered higher SPT phases. These higher SPT's are protected by a symmetry $d$-group, a higher categorical version of a group which can charge not just particles but also strings, volumes, etc. This structure emerges naturally from computation of the anomaly. One can interpret these severe anomalies as producing an emergent higher-form gauge field when $G$ is gauged, and this gauge field participates in the bulk topological action.

Figures (1)

• Figure 1: An electric charge encircles a codimension 2 defect where two domain walls $x, y$ fuse to a third.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Theorem 3