Dressing and Screening in Anti-de Sitter

TL;DR

This work analyzes abelian gauge theory with matter in AdS, revealing how Dirichlet boundary data can induce a one-loop photon mass via a border-triggered Higgs-like mechanism and how Neumann boundary conditions require geodesic-Wilson-line dressing to construct gauge-invariant observables. The authors perform explicit one-loop computations of the photon self-energy and four-point functions, derive a Ward-identity-based derivation of the Higgs mechanism, and show that the boundary tilt operator encodes the spontaneous breaking of a bulk one-form symmetry with a protected boundary current. A central theme is the interpretation of gauge dynamics in terms of generalized symmetries, including a nonperturbative argument for the decoupling of boundary field strengths when Neumann conditions are used, grounded in higher-form symmetry. The results illuminate how bulk gauge invariance and boundary conformal structure interplay in AdS, with potential applications to de Sitter space via analytic continuation and to constructing gauge-invariant observables in cosmology and holography.

Abstract

Motivated by a question of defining gauge-invariant observables in cosmology and by the close connection between perturbation theory in de Sitter (dS) and Anti-de Sitter (AdS), we study scalar electrodynamics in AdS in setups that are largely unexplored but relevant for dS physics. For photons with standard (Dirichlet) boundary conditions, we analyze charged scalars whose boundary conditions break the $U(1)$ symmetry. This leads to a nonstandard Higgs mechanism in which the gauge field acquires a one-loop mass without a classical vacuum expectation value. Using recent advances in perturbation theory in AdS, we compute this mass explicitly and evaluate charged-scalar four-point functions. We also provide an alternative derivation based on boundary Ward identities. For photons with alternate (Neumann) boundary conditions, where local charged operators are not gauge invariant, we construct physical observables by dressing charged fields with geodesic Wilson lines. These dressed operators have well-behaved conformal properties and unphysical photon modes decouple from their correlation functions. Explicit one-loop computations further reveal the decoupling of boundary field strengths, for which we provide a nonperturbative argument based on higher-form symmetry. Along the way, we explain the physical consequences of spontaneous breaking of higher-form symmetry in AdS, including the role of the tilt operator, the relation between one-form symmetry and endpoints of Wilson lines at the boundary, and a generalized-symmetry interpretation of conserved currents dual to bulk gauge fields.

Figures (17)

• Figure 1: Witten diagram representing the photon exchange. This diagram is gauge invariant for the exchange of a gauge field with Dirichlet boundary condition, but it is not for a Neumann exchange.
• Figure 2: Witten diagrams representing the geodesic-photon-geodesic exchange on the left and the geodesic-photon-current exchange on the right; the geodesics are depicted in green, and the exchanged photon is intended with Neumann boundary conditions.
• Figure 3: The geodesic-photon-geodesic diagram factorizes into the product of CFT four-point function of the matter insertions, and a photon exchange integrated along geodesics.
• Figure 4: The four-point function of the matter fields must be unit normalized to be consistent with the normalization of the bulk-to-boundary propagator adopted in the rest of the document.
• Figure 5: Adopting the split representation of spin-1 harmonic function, it becomes necessary to compute the spin-1 propagator from a boundary point $P_5$ to a point integrated over the geodesic connecting $P_1$ and $P_2$.