Source and Response Soft Charges for Maxwell Theory on $AdS_d$

TL;DR

This paper investigates asymptotic symmetries and their charges in Maxwell theory on AdS backgrounds, revealing an infinite-dimensional algebra of soft charges and their behavior in the flat space limit, with implications for AdS/CFT.

Contribution

It constructs a conserved symplectic structure for Maxwell theory on AdS and identifies the boundary phase space with scalar fields and gauge transformations, revealing an infinite-dimensional Heisenberg algebra of charges.

Findings

Boundary phase space described by two scalars and gauge transformations.

Soft charges form an infinite-dimensional Heisenberg algebra.

In the flat space limit, only source soft charges remain.

Abstract

We study asymptotic symmetries and their associated charges for Maxwell theory on anti de Sitter (AdS) background in any dimension. This is obtained by constructing a conserved symplectic structure for the bulk and a theory on the boundary, which we specify. We show that the boundary phase space is described by two scalars and two sets of "source" and "response" boundary gauge transformations. The bulk dynamics is invariant under these two sets of boundary transformations. We study the (soft) charges associated with these two sets and show that they form an infinite dimensional Heisenberg type algebra. Studying the large AdS radius flat space limit, we show only the source soft charges survive. We also analyze algebra of charges associated with SO(d-1,2) isometries of the background $AdS_d$ space and study how they act on our source and response charges. We briefly discuss implication of our results for the AdS/CFT.