Anomalous Symmetries End at the Boundary

TL;DR

This paper demonstrates that 't Hooft anomalies in quantum field theories prevent the consistent definition of symmetric boundary conditions, extending the understanding of anomalies to boundary phenomena and various symmetry types.

Contribution

It establishes that anomalies serve as obstructions to symmetric boundary conditions, including for gravitational and discrete symmetries, and proposes a conjecture for gravitational anomalies.

Findings

Anomalies obstruct symmetric boundary conditions in QFTs.

The analysis applies to Lorentz, gravitational, and discrete symmetries.

The results hold for theories in up to four dimensions.

Abstract

A global symmetry of a quantum field theory is said to have an 't Hooft anomaly if it cannot be promoted to a local symmetry of a gauged theory. In this paper, we show that the anomaly is also an obstruction to defining symmetric boundary conditions. This applies to Lorentz symmetries with gravitational anomalies as well. For theories with perturbative anomalies, we demonstrate the obstruction by analyzing the Wess-Zumino consistency conditions and current Ward identities in the presence of a boundary. We then recast the problem in terms of symmetry defects and find the same conclusions for anomalies of discrete and orientation-reversing global symmetries, up to the conjecture that global gravitational anomalies, which may not be associated with any diffeomorphism symmetry, also forbid the existence of boundary conditions. This conjecture holds for known gravitational anomalies in $D \le 3$ which allows us to conclude the obstruction result for $D \le 4$.