Gauge-invariant observables, gravitational dressings, and holography in AdS

TL;DR

The paper constructs gauge-invariant bulk observables in AdS by gravitationally dressing local operators at leading order in the gravitational coupling, illustrating line and Coulomb dressings that make the operators nonlocal. It derives explicit boundary charges and a boundary stress-tensor interpretation of AdS isometries, and analyzes how these dressings affect holographic ideas, including extrapolate maps and HKLL-type reconstructions. A central result is that while gravity provides a boundary Hamiltonian through nonlocal observables, a fully consistent holographic map and unitary bulk evolution likely require solving the nonperturbative bulk constraint equations, highlighting limits of perturbative holography. The work clarifies how gravitational dressing influences bulk-boundary relations and emphasizes the role of boundary data in encoding bulk dynamics, with implications for understanding holography in AdS beyond leading order.

Abstract

This paper explores construction of gauge (diffeomorphism)-invariant observables in anti de Sitter (AdS) space and the related question of how to find a "holographic map" providing a quantum equivalence to a boundary theory. Observables are constructed perturbatively to leading order in the gravitational coupling by gravitationally dressing local field theory operators in order to solve the gravitational constraints. Many such dressings are allowed and two are explicitly examined, corresponding to a gravitational line and to a Coulomb field; these also reveal an apparent role for more general boundary conditions than considered previously. The observables obey a nonlocal algebra, and we derive explicit expressions for the boundary generators of the SO(D-1,2) AdS isometries that act on them. We examine arguments that gravity {\it explains} holography through the role of such a boundary Hamiltonian. Our leading-order gravitational construction reveals some questions regarding how these arguments work, and indeed construction of such a holographic map appears to require solution of the non-perturbative generalization of the bulk constraint equations.