4d Maxwell on the Edge: Global Aspects of Boundary Conditions and Duality

TL;DR

This paper explores the global boundary conditions of 4d Maxwell theory, analyzing boundary line operators, dualities, and symmetries, and introduces a 5d SymTFT perspective to unify these aspects.

Contribution

It provides a comprehensive analysis of boundary conditions, dualities, and symmetries in 4d Maxwell theory, including a novel 5d SymTFT framework for understanding these features.

Findings

Identification of boundary Wilson-'t Hooft lines and their topological properties

Mapping boundary conditions via 3d topological interfaces implementing dualities

Derivation of an $SL(2, ext{Q})$ action on the bulk coupling $ au$

Abstract

We revisit Maxwell theory in 4d with a boundary, with particular attention to the global properties of the boundary conditions, both in the free (topological) and interacting (conformal) cases. We analyze the fate of Wilson-'t Hooft lines, identifying the subset that is trivialized on the boundary and the ones that become topological, thus generating a boundary 1-form symmetry. We further study how the boundary conditions are mapped to each other by 3d topological interfaces implementing bulk dualities and rescalings of the coupling. Together, these interfaces generate an $SL(2,\mathbb{Q})$ action on the bulk complexified coupling $\tau$, and they generalize the usual $SL(2,\mathbb{Z})$ action on 3d CFTs by including both topological and non-topological manipulations within a unified framework. We then show how to recover our results in a streamlined way from a SymTFT picture in 5d with corners. Finally, we comment on the possible inclusion of non-compact 3d edge modes.