$p$-Form Surface Charges on AdS: Renormalization and Conservation
Erfan Esmaeili, Vahid Hosseinzadeh
TL;DR
The paper tackles the problem of defining and renormalizing surface charges for $p$-form gauge fields on the boundary of $AdS_{d+1}$, clarifying how holographic counter-terms and corner terms shape the finite symplectic structure. It develops a near-boundary analysis, decomposing the $(p+1)$-form into spherical harmonics to obtain gauge-invariant variables and the renormalized action, then studies how different boundary conditions (including conservative and leaky cases denoted by $oldsymbol{F}$, $oldsymbol{ riangle}$, and $oldsymbol{oxempty}$) yield distinct charge and flux expressions. The work demonstrates that leaky boundary conditions support infinite-dimensional Abelian charge algebras, identifies conditions for conserved charges in a phase-space subset, and clarifies the role of electric/magnetic duality in defining magnetic charges. It also exposes nontrivial issues of integrability of energy due to boundary gauge modes, despite a universal bulk energy, and discusses dual formulations that map magnetic charges to electric charges of a dual theory. The results advance the holographic understanding of higher-form symmetries in AdS/CFT and illuminate how corner terms and dualities influence observable boundary charges in holographic settings.
Abstract
Surface charges of a $p$-form theory on the boundary of an AdS$_{d+1}$ spacetime are computed. Counter-terms on the boundary produce divergent corner-terms which holographically renormalize the symplectic form. Different choices of boundary conditions lead to various expressions for the charges and the associated fluxes. With the usual standard AdS boundary conditions, there are conserved zero-mode charges. Moreover, we explore two leaky boundary conditions which admit an infinite number of charges forming an Abelian algebra and non-vanishing flux. Finally, we discuss magnetic $p$-form charges and electric/magnetic duality.