Poincare-Einstein Holography for Forms via Conformal Geometry in the Bulk

TL;DR

This paper develops a conformally invariant calculus for higher form equations on Einstein manifolds with boundary, providing explicit holographic formulas for conformal operators and solutions, advancing the understanding of Poincare-Einstein holography.

Contribution

It introduces a new tractor calculus framework for solving higher form Proca equations and deriving explicit boundary operators in conformal geometry.

Findings

Constructed an operator projecting forms to solutions of boundary problems.

Developed a product formula for asymptotic solutions in conformal geometry.

Derived explicit holographic formulas for conformal operators on differential forms.

Abstract

We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic problems in general. The central tools of our approach are (i) the conformal geometry of differential forms and the associated exterior tractor calculus, and (ii) a generalised notion of scale which encodes the connection between the underlying geometry and its boundary. The latter also controls the breaking of conformal invariance in a very strict way by coupling conformally invariant equations to the scale tractor associated with the generalised scale. From this, we obtain a map from existing solutions to new ones that exchanges Dirichlet and Neumann boundary conditions. Together, the scale tractor and exterior structure extend the solution generating algebra of [31] to a conformally invariant, Poincare--Einstein calculus on (tractor) differential forms. This calculus leads to explicit holographic formulae for all the higher order conformal operators on weighted differential forms, differential complexes, and Q-operators of [9]. This complements the results of Aubry and Guillarmou [3] where associated conformal harmonic spaces parametrise smooth solutions.