Anomaly Obstructions to Symmetry Preserving Gapped Phases

TL;DR

This paper demonstrates that certain anomalies in discrete symmetries impose strict constraints on the phases of quantum field theories, preventing the existence of certain gapped, symmetry-preserving states.

Contribution

It introduces an anomaly inflow obstruction criterion that determines when a symmetry-preserving gapped phase can exist in theories with discrete symmetries.

Findings

Obstruction must vanish for a gapped, symmetry-preserving phase to exist.

Certain 4d gauge theories at θ=π cannot be gapped and symmetric.

Constraints on 4d adjoint QCD dynamics.

Abstract

Anomalies are renormalization group invariants that constrain the dynamics of quantum field theories. We show that certain anomalies for discrete global symmetries imply that the underlying theory either spontaneously breaks its generalized global symmetry or is gapless. We identify an obstruction, formulated in terms of the anomaly inflow action, that must vanish if a symmetry preserving gapped phase, i.e. a unitary topological quantum field theory, exits with the given anomaly. Our result is similar to the $2d$ Lieb-Schultz-Mattis theorem but applies more broadly to continuum theories in general spacetime dimension with various types of discrete symmetries including higher-form global symmetries. As a particular application, we use our result to prove that certain $4d$ non-abelian gauge theories at $\theta=\pi$ cannot flow at long distances to a phase which simultaneously, preserves time-reversal symmetry, is confining, and is gapped. We also apply our obstruction to $4d$ adjoint QCD and constrain its dynamics.