ABJ anomaly as a U(1) symmetry and Noether's theorem

TL;DR

This paper explores the Adler-Bell-Jackiw anomaly as a U(1) symmetry, revealing its role in transforming non-local sectors and providing insights into anomaly quantization, matching, and the nature of Noether currents.

Contribution

It introduces a novel perspective on the ABJ anomaly as a U(1) symmetry that acts on non-local operator sectors, unifying various aspects of anomaly theory.

Findings

The U(1) symmetry transforms Haag duality violating sectors.

Provides a unifying view on anomaly quantization and matching.

Discusses the implications for Noether's theorem and non-invertible symmetries.

Abstract

The Adler-Bell-Jackiw anomaly determines the violation of chiral symmetry when massless fermions are coupled to an abelian gauge field. In its seminal paper, Adler noticed that a modified chiral U(1) symmetry could still be defined, at the expense of being generated by a non-gauge-invariant conserved current. We show this internal U(1) symmetry has the special feature that it transforms the Haag duality violating sectors (or non-local operator classes). This provides a simple unifying perspective on the origin of anomaly quantization, anomaly matching, applicability of Goldstone theorem, and the absence of a Noether current. We comment on recent literature where this symmetry is considered to be either absent or non-invertible. We end by recalling the DHR reconstruction theorem, which states 0-form symmetries cannot be non-invertible for d>2, and argue for a higher-form symmetry reconstruction theorem.