Non-invertible self-duality defects of Cardy-Rabinovici model and mixed gravitational anomaly

TL;DR

This paper investigates the non-invertible self-duality symmetry in the 4d Cardy-Rabinovici model, revealing its complex realization, associated anomalies, and implications for the phase diagram of the theory.

Contribution

It demonstrates that the self-duality in the model is non-invertible due to fusion rule complexities and identifies a mixed gravitational anomaly affecting the phase structure.

Findings

Self-duality is realized as a non-invertible symmetry.

A mixed gravitational anomaly is detected on K3 surfaces.

Trivially gapped phases are ruled out by anomaly matching.

Abstract

We study properties of self-duality symmetry in the Cardy-Rabinovici model. The Cardy-Rabinovici model is the $4$d $U(1)$ gauge theory with electric and magnetic matters, and it enjoys the $SL(2,\mathbb{Z})$ self-duality at low-energies. $SL(2,\mathbb{Z})$ self-duality does not realize in a naive way, but we notice that the $ST^{p}$ duality transformation becomes the legitimate duality operation by performing the gauging of $\mathbb{Z}_N$ $1$-form symmetry with including the level-$p$ discrete topological term. Due to such complications in its realization, the fusion rule of duality defects becomes a non-group-like structure, and thus the self-duality symmetry is realized as a non-invertible symmetry. Moreover, for some fixed points of the self-duality, the duality symmetry turns out to have a mixed gravitational anomaly detected on a $K3$ surface, and we can rule out the trivially gapped phase as a consequence of anomaly matching. We also uncover how the conjectured phase diagram of the Cardy-Rabinovici model satisfies this new anomaly matching condition.