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Poincaré–Einstein holography for forms via conformal geometry in the bulk


About this Title

A. Rod Gover, Emanuele Latini and Andrew Waldron

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 235, Number 1106
ISBNs: 978-1-4704-1092-6 (print); 978-1-4704-2224-0 (online)
DOI: http://dx.doi.org/10.1090/memo/1106
Published electronically: September 23, 2014
Keywords:Differential forms, conformally compact, Q-curvature, conformal harmonics, Poincaré–Einstein, AdS/CFT, holography, scattering

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Bulk conformal geometry and extension problems
  • Chapter 3. Tractor exterior calculus
  • Chapter 4. The exterior calculus of scale
  • Chapter 5. Higher form Proca equations
  • Chapter 6. Obstructions, detours, gauge operators and $Q$-curvature
  • Appendix A. The ambient manifold
  • Appendix B. List of common symbols

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 Gover and Waldron to a conformally invariant, Poincaré–Einstein calculus on (tractor) differential forms. This calculus leads to explicit holographic formulæfor all the higher order conformal operators on weighted differential forms, differential complexes, and Q-operators of Branson and Gover (2005). This complements the results of Aubry and Guillarmou where associated conformal harmonic spaces parametrise smooth solutions.

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