The mixed $0$-form/$1$-form anomaly in Hilbert space: pouring new wine into old bottles

TL;DR

This paper investigates the mixed anomaly between 0-form and 1-form symmetries in 4D gauge theories, revealing how it leads to spectrum degeneracies and constrains the theories' infrared behavior.

Contribution

It provides a canonical quantization framework on a torus with background fields, determining the global symmetry algebra and its anomalies in these theories.

Findings

Mixed anomalies cause spectrum degeneracies at finite volume.

The global symmetry algebra is centrally extended due to the anomaly.

Results are consistent with semiclassical and infrared phase analyses.

Abstract

We study four-dimensional gauge theories with arbitrary simple gauge group with $1$-form global center symmetry and $0$-form parity or discrete chiral symmetry. We canonically quantize on $\mathbb{T}^3$, in a fixed background field gauging the $1$-form symmetry. We show that the mixed $0$-form/$1$-form 't Hooft anomaly results in a central extension of the global-symmetry operator algebra. We determine this algebra in each case and show that the anomaly implies degeneracies in the spectrum of the Hamiltonian at any finite-size torus. We discuss the consistency of these constraints with both older and recent semiclassical calculations in $SU(N)$ theories, with or without adjoint fermions, as well as with their conjectured infrared phases.