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

TL;DR

The paper develops a framework in which the Cardy-Rabinovici 4d U(1) gauge theory hosts non-invertible self-duality defects generated by gauging a $\mathbb{Z}_N^{[1]}$ 1-form symmetry with a level-$p$ discrete theta term. Focusing on the ST^{-1} fixed point, it constructs a codimension-1 non-invertible defect via half-space gauging and derives its non-group-like fusion rules, revealing a mixed gravitational anomaly on K3 that constrains infrared dynamics. The authors show that trivially gapped phases cannot realize the ST^{-1} duality at $\tau_* = e^{\pi i/3}$ and propose an anomaly-matching condition that ties together the Higgs, monopole-confinement, and dyon-confinement phases, consistent with the conjectured phase diagram. They also discuss broader implications for duality defects in 4d gauge theories and potential generalizations to other dualities, including connections to $\mathcal{N}=4$ SYM and beyond.

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.

Figures (3)

• Figure 1: Conjectured phase diagram of the Cardy-Rabinovici model. In the weak-coupling regime, electric charge condensation occurs and the system is in the Higgs phase. In the strong-coupling regime, magnetic charge condensation occurs. Depending on the values of $\theta$, different types of dyon start to condense, and the Cardy-Rabinovici model is expected to have the rich phase structure.
• Figure 2: Fusion rule of the duality defect $\mathscr{D}(M^{(3)})$ and its orientation reverse $\Bar{\mathscr{D}}(M^{(3)})$. We infinitesimally displace those hypersurfaces as $\mathscr{D}(M^{(3)}_{+\varepsilon})$ and $\Bar{\mathscr{D}}(M^{(3)}_{-\varepsilon})$.
• Figure 3: Fusion rule of three duality defects $\mathscr{D}(M^{(3)})$, and those hypersurfaces are infinitesimally displaced as $\mathscr{D}(M^{(3)}_1)$, $\mathscr{D}(M^{(3)}_2)$, and $\mathscr{D}(M^{(3)}_3)$.

Theorems & Definitions (4)

• Claim 1
• Claim 2: Corollary of Claim \ref{['claim:self-duality-1']}
• Claim 3
• Claim 4: Mixed gravitational anomaly matching