Defect Charges, Gapped Boundary Conditions, and the Symmetry TFT

TL;DR

This paper introduces a new framework using Symmetry TFT to characterize higher charges of defect operators, linking them to gapped boundary conditions and exploring applications in anomalous theories and duality symmetries.

Contribution

It provides a streamlined, computational approach to classify higher defect charges via boundary conditions in Symmetry TFT, generalizing anomaly-boundary relations.

Findings

Charges correspond to gapped boundary conditions on specific manifolds.

The framework applies to anomalous bulk theories hosting symmetric defects.

Properties of surface charges in 3+1d duality symmetries are described.

Abstract

We offer a streamlined and computationally powerful characterization of higher representations (higher charges) for defect operators under generalized symmetries, employing the powerful framework of Symmetry TFT $\mathcal{Z}(\mathcal{C})$. For a defect $\mathscr{D}$ of codimension p, these representations (charges) are in one-to-one correspondence with gapped boundary conditions for the SymTFT $\mathcal{Z}(\mathcal{C})$ on a manifold $Y = \Sigma_{d-p+1} \times S^{p-1}$, and can be efficiently described through dimensional reduction. We explore numerous applications of our construction, including scenarios where an anomalous bulk theory can host a symmetric defect. This generalizes the connection between 't Hooft anomalies and the absence of symmetric boundary conditions to defects of any codimension. Finally we describe some properties of surface charges for (3 + 1)d duality symmetries, which should be relevant to the study of Gukov-Witten operators in gauge theories.