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

TL;DR

This work classifies and interrelates Maxwell boundary conditions in 4d using a 3d topological edge-mode framework and a 5d SymTFT (SymTFT) perspective. It shows how free boundary data $(P,Q,\tilde r)$ determine the fate of Wilson and 't Hooft lines and how topological interfaces implement bulk dualities, yielding an $SL(2,\mathbb{Q})$ action on the coupling $\tau$ in addition to the standard $SL(2,\mathbb{Z})$ action on 3d CFTs in the interacting case. The interacting sector couples 3d CFTs to the bulk with boundary data $B^{(p,v,k,u)}(\tau,\overline{\tau})$, and the SymTFT formalism encodes all dualities and boundary maps via oblique compactifications and corner interfaces. The paper also discusses non-compact edge modes, condensation defects, and potential generalizations to non-Abelian theories, providing a cohesive, geometrical toolkit for global boundary conditions and dualities in Maxwell theory. These results sharpen the understanding of boundary/global structures in gauge theories and offer a robust route to exploring holographic and edge-mode phenomena in related QFTs.

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 $τ$, 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.

Figures (5)

• Figure 1: Acting with a topological interface we can relate a boundary condition with data $(p,v,k)$ in the theory at coupling $\tau$ to a new one with data $(p_f,v_f,k_f)$ in the theory at coupling $\tau'$.
• Figure 2: Top: How bulk lines ending on a particular boundary condition (gray) can become non-endable after the action of a condensation defect (green). Bottom: Similarly, bulk lines trivialized by the same boundary conditions become non-trivial, but topological (dashed red) after acting with the condensation defect.
• Figure 3: Two possible ways of constructing a Boundary SymTFT. Left: The physical 3d boundary serves as a corner between the non-topological and topological boundary conditions of the slab. Right: A new 3d topological corner is introduced as an interface between two 4d topological boundaries $\mathcal{B}_{4d ,top.}$ and $\mathcal{B}_{4d' ,top.}$. The choice of this new data determines the topological couplings of the physical 3d boundary condition.
• Figure 4: Oblique slab compactification of the 5d theory on an interval. On the left, the full 5d/4d/3d setup with physical and topological boundaries. On the right, the phyiscal 4d theory with a topological interface between $\tau$ and $\tau'$ and a boundary.
• Figure 5: Illustration of the SL$(2,\mathbb{Z})$ 0-form symmetry, realized as a codimension-one defect of the 5d TFT, acting on the topological boundaries. In (a) it transforms the $4d'$ boundary, shifting $\tau'$ and leaving $4d$ fixed; in (b) it acts on $4d$, leaving $4d'$ unchanged. In both cases, it induces an action on the 3d topological corner.