Wedge Domains in Compactly Causal Symmetric Spaces

Neeb KH, Olafsson G (2022)

Publication Type: Journal article

Publication year: 2022

Journal

International Mathematics Research Notices Oxford University Press (OUP): Policy H - Oxford Open Option A

Abstract

Motivated by constructions in Algebraic Quantum Field Theory we introduce wedge domains in compactly causal symmetric spaces M=G/H, which includes in particular anti-de Sitter space in all dimensions and its coverings. Our wedge domains generalize Rindler wedges in Minkowski space. The key geometric structure we use is the modular flow on M defined by an Euler element in the Lie algebra of G. Our main geometric result asserts that three seemingly different characterizations of these domains coincide: the positivity domain of the modular vector field, the domain specified by a KMS-like analytic extension condition for the modular flow, and the domain specified by a polar decomposition in terms of certain cones. In the second half of the article we show that our wedge domains share important properties with wedge domains in Minkowski space. If G is semisimple, there exist unitary representations (U, H) of G and isotone covariant nets of real subspaces H(O) subset of H, defined for any open subset O subset of M, which assign to connected components of the wedge domains a standard subspace whose modular group corresponds to the modular f low on M. This corresponds to the Bisognano-Wichmann property in Quantum Field Theory. We also show that the set of G-translates of the connected components of the wedge domain provides a geometric realization of the abstract wedge space introduced by the first author and V. Morinelli.

Authors with CRIS profile

Karl Hermann Neeb Lehrstuhl für Mathematik (Lie-Gruppen und Darstellungstheorie)

Involved external institutions

Louisiana State University

United States (USA) (US)

How to cite

APA:

Neeb, K.H., & Olafsson, G. (2022). Wedge Domains in Compactly Causal Symmetric Spaces. International Mathematics Research Notices. https://dx.doi.org/10.1093/imrn/rnac131

MLA:

Neeb, Karl Hermann, and Gestur Olafsson. "Wedge Domains in Compactly Causal Symmetric Spaces." International Mathematics Research Notices (2022).

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