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

TL;DR

The paper develops a unified obstruction framework for anomalies of non-invertible self-duality symmetries in 2d and 4d using the Symmetry TFT. It identifies two key obstructions to gauging duality defects: (i) the existence of a $G$-invariant Lagrangian algebra in the bulk Dijkgraaf–Witten theory, and (ii) a residual pure ‘t Hooft anomaly controlled by the equivariantization data that encodes symmetry fractionalization. In 2d, this reproduces the Tambara–Yamagami anomaly structure through a bulk perspective and clarifies the role of the Arf invariant of the fractionalization class; in 4d, it generalizes to duality and triality defects with a 5d bulk, giving a compact criterion for anomaly by a mixed 1-form/0-form symmetry interplay. The results yield concrete constraints for RG flows and boundary conditions in self-dual theories, with implications for $\mathcal{N}=3$ theories and gravity couplings, and offer a pathway to analyze higher-dimensional non-invertible symmetries via symmetry TFT and equivariantization data.

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.

Figures (2)

• Figure 1: Graphical representation of the action of a symmetry defect $U_g$ on the junction spaces $V_{x ,\, y}^z$ (above) and on the projectors $\pi_x$ (below).
• Figure 2: Braiding of lines $W^L_a$ and $W^R_b$ on the duality defect. Unlinking the line configuration gives rise to the symmetric bicharacter $\gamma(a,b)$.