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

We study new symmetries of the Cardy-Rabinovici model and their dynamical applications. The Cardy-Rabinovici model is a 4d U (1) gauge theory with electric and magnetic matters, which is a good playground for studying the dynamics of the Yang-Mills theory with θ angle. In this model, the electromagnetic S L (2, ℤ) self-duality is not realized in a naive way. Still, the S L (2, ℤ) transformations become legitimate duality operations by appropriately gauging the ℤ N 1-form symmetry. We construct new noninvertible symmetries from this duality at self-dual points and determine their non-group-like fusion rules. As an application, we can rule out the trivially gapped phase for some self-dual parameters due to a mixed gravitational anomaly of this new symmetry. We also show how the conjectured phase diagram of the Cardy-Rabinovici model is consistent with this anomaly matching condition.