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Letters in Mathematical Physics

, Volume 107, Issue 12, pp 2291–2331 | Cite as

The holographic Hadamard condition on asymptotically anti-de Sitter spacetimes

  • Michał WrochnaEmail author
Article

Abstract

In the setting of asymptotically anti-de Sitter spacetimes, we consider Klein–Gordon fields subject to Dirichlet boundary conditions, with mass satisfying the Breitenlohner–Freedman bound. We introduce a condition on the \(\mathrm{b}\)-wave front set of two-point functions of quantum fields, which locally in the bulk amounts to the usual Hadamard condition, and which moreover allows to estimate wave front sets for the holographically induced theory on the boundary. We prove the existence of two-point functions satisfying this condition and show their uniqueness modulo terms that have smooth Schwartz kernel in the bulk and have smooth restriction to the boundary. Finally, using Vasy’s propagation of singularities theorem, we prove an analogue of Duistermaat and Hörmander’s theorem on distinguished parametrices.

Keywords

Quantum field theory on curved spacetimes Asymptotically anti-de Sitter spacetimes Holography Hadamard condition 

Mathematics Subject Classification

81T13 81T20 35S05 35S35 

Notes

Acknowledgements

The author is deeply grateful to András Vasy for all the discussions and helpful comments. The author would also like to thank Claudio Dappiaggi, Christian Gérard, Peter Hintz, Jacques Smulevici and Jochen Zahn for stimulating discussions and useful remarks. Financial support from the ANR-16-CE40-0012-01 grant is gratefully acknowledged. The author is also grateful to the Erwin Schrödinger Institute in Vienna for its hospitality during the program “Modern theory of wave equations”.

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Authors and Affiliations

  1. 1.Institut Fourier, UMR 5582 CNRSUniversité Grenoble AlpesGrenoble Cedex 09France

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