Anomalies of non-invertible self-duality symmetries: fractionalization and gauging

TL;DR

This paper investigates anomalies in non-invertible duality symmetries in 2d and 4d using the Symmetry TFT, revealing obstructions to gauging and linking anomalies to symmetry fractionalization and Lagrangian algebras.

Contribution

It generalizes the obstruction theory for Tambara-Yamagami categories to higher dimensions and characterizes anomalies in duality symmetries through symmetry fractionalization.

Findings

Identified two obstructions to gauging duality defects.

Linked anomaly conditions to symmetry fractionalization.

Provided a compact characterization of anomalies in 4d theories.

Abstract

We study anomalies of non-invertible duality symmetries in both 2d and 4d, employing the tool of the Symmetry TFT. In the 2d case we rephrase the known obstruction theory for the Tambara-Yamagami fusion category in a way easily generalizable to higher dimensions. In both cases we find two obstructions to gauging duality defects. The first obstruction requires the existence of a duality-invariant Lagrangian algebra in a certain Dijkgraaf-Witten theory in one dimension more. In particular, intrinsically non-invertible (a.k.a. group theoretical) duality symmetries are necessarily anomalous. The second obstruction requires the vanishing of a pure anomaly for the invertible duality symmetry. This however depends on further data. In 2d this is specified by a choice of equivariantization for the duality-invariant Lagrangian algebra. We propose and verify that this is equivalent to a choice of symmetry fractionalization for the invertible duality symmetry. The latter formulation has a natural generalization to 4d and allows us to give a compact characterization of the anomaly. We comment on various possible applications of our results to self-dual theories.