Soft edges: the many links between soft and edge modes

Abstract: Boundaries in gauge theory and gravity give rise to symmetries and charges at both finite and asymptotic distance. Due to their structural similarities, it is often held that soft modes are some kind of asymptotic limit of edge modes. Here, we show in Maxwell theory that there is an arguably more interesting relationship between the asymptotic symmetries and their charges, on one hand, and their finite-distance counterparts, on the other, without the need of a limit. Key to this observation is to embed the finite region in the global spacetime and identify edge modes as dynamical $\rm{U}(1)$-reference frames for dressing subregion variables. Distinguishing intrinsic and extrinsic frames, according to whether they are built from field content in- or outside the region, we find that non-trivial corner symmetries arise only for extrinsic frames. Further, the asymptotic-to-finite relation requires asymptotically charged ones (like Wilson lines). Such frames, called soft edges, extend to asymptotia and, in fact, realize the corner charge algebra in multiple ways, for example, by "pulling in" the asymptotic one from infinity, or physically through the addition of asymptotic soft and hard radiation. Realizing an infinite-dimensional algebra requires a new set of soft boundary conditions, relying on the distinction between extrinsic and intrinsic data. We identify the subregion Goldstone mode as the relational observable between extrinsic and intrinsic frames and clarify the meaning of vacuum degeneracy. We also connect the asymptotic memory effect with a more operational quasi-local one. A main conclusion is that the relationship between asymptotia and finite distance is frame-dependent; each choice of soft edge mode probes distinct cross-boundary data of the global theory.