Renormalization of spin-one asymptotic charges in AdS$_D$

TL;DR

This work develops a systematic renormalization framework for Maxwell fields in AdS$_D$ to define finite boundary charges for angle-dependent asymptotic symmetries. By performing holographic renormalization in Poincaré coordinates and covariant symplectic renormalization, the authors derive finite, coordinate-invariant charges in both even and odd dimensions, including explicit treatments in Bondi coordinates that accommodate a smooth flat limit. They provide general counterterm structures and corner-term prescriptions, show how charges depend on boundary data (sources and Coulombic/Vev data), and demonstrate consistency between Poincaré and Bondi formulations as well as with the $ ext{AdS}$ to flat transition. The results illuminate how infinite-dimensional asymptotic symmetries in higher dimensions can be realized with finite charges, and offer a concrete route to extend to other linear gauge theories and higher-spin fields. The formalism ties holographic renormalization to the covariant phase space approach, enabling robust definitions of charges and their flux across $ ext{AdS}$ boundaries and at null infinity in the flat limit.$

Abstract

We study the renormalized action and the renormalized presymplectic potential for Maxwell fields on Anti de Sitter backgrounds of any dimensions. We then use these results to explicitly derive finite boundary charges for angle-dependent asymptotic symmetries. We consider both Poincaré and Bondi coordinates, the former allowing us to control the systematics for arbitrary $D$, the latter being better suited for a smooth flat limit.

Figures (2)

• Figure 1: A representation of AdS$_2$. The red line is the $R=0$ submanifold, the solid green line is the one at $T=0$ and the dashed green line is the one at $T=\pi\ell$. The purple lines mark the intersection with the plane $X^1 = X^2$: the portion of the spacetime below this plane is covered by the Poincaré coordinates, see Eq. \ref{['PoincareRange']}.
• Figure 2: Penrose diagrams for AdS$_2$ and AdS$_3$. The highlighted region is the portion of spacetime covered by the Poincaré coordinate patch (see Eq. \ref{['PoincareRange']}). The boundary of this region has two components (see below Eq. \ref{['PoincareRange']}): $z=0$, which lies on the actual boundary of AdS$_D$, and $z=\infty$, which lies in the interior of AdS$_D$.