Higher Structure of Chiral Symmetry

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.