Higher Structure of Chiral Symmetry

TL;DR

The authors develop a higher-categorical framework for non-invertible symmetries in $D$-dimensional QFTs, where symmetry data are encoded in a fusion $(D-1)$-category and associativity is captured by $F$-symbols valued in $(D-2)$-dimensional TFTs. They instantiate this in the non-invertible chiral symmetry of four-dimensional massless QED, describing defects $oldsymbol{rak D}_{p/N} = {uildrel ext{χ} elax angle}_{p/N} \,oxtimes\ ext{A}^{N,p}$ whose fusion channels carry TFT data and whose 1-morphisms implement gauging interfaces. They show that the associated F-symbol TFTs can be probed via correlators with extended operators and through Ward identities on non-trivial four-manifolds obtained by surgery, all without relying on a Lagrangian description. The work reveals a structured path toward a Symmetry TFT perspective for general non-invertible higher symmetries and suggests broad future applications to higher-dimensional theories and axion electrodynamics.

Abstract

A recent development in our understanding of the theory of quantum fields is the fact that familiar gauge theories in spacetime dimensions greater than two can have non-invertible symmetries generated by topological defects. The hallmark of these non-invertible symmetries is that the fusion rule deviates from the usual group-like structure, and in particular the fusion coefficients take values in topological field theories (TFTs) rather than in mere numbers. In this paper we begin an exploration of the associativity structure of non-invertible symmetries in higher dimensions. The first layer of associativity is captured by F-symbols, which we find to assume values in TFTs that have one dimension lower than that of the defect. We undertake an explicit analysis of the F-symbols for the non-invertible chiral symmetry that is preserved by the massless QED and explore their physical implications. In particular, we show the F-symbol TFTs can be detected by probing the correlators of topological defects with 't Hooft lines. Furthermore, we derive the Ward-Takahashi identity that arises from the chiral symmetry on a large class of four-dimensional manifolds with non-trivial topologies directly from the topological data of the symmetry defects, without referring to a Lagrangian formulation of the theory.