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Kinematic space and the orbit method

  • Robert F. PennaEmail author
  • Claire Zukowski
Open Access
Regular Article - Theoretical Physics
  • 40 Downloads

Abstract

Kinematic space has been defined as the space of codimension-2 spacelike extremal surfaces in anti de Sitter (AdSd+1) spacetime which, by the Ryu-Takayanagi proposal, compute the entanglement entropy of spheres in the boundary CFTd. It has recently found many applications in holography. Coadjoint orbits are symplectic manifolds that are the classical analogues of a Lie group’s unitary irreducible representations. We prove that kinematic space is a particular coadjoint orbit of the d-dimensional conformal group SO(d, 2). In addition, we show that the Crofton form on kinematic space associated to AdS3, that was shown to compute the lengths of bulk curves, is equal to the standard Kirillov-Kostant symplectic form on the coadjoint orbit. Since kinematic space is Kähler in addition to symplectic, it can be quantized. The orbit method extends the kinematic space dictionary, which was originally motivated through connections to integral geometry, by directly translating geometrical properties of holographic auxiliary spaces into statements about the representation theory of the conformal group.

Keywords

AdS-CFT Correspondence Black Holes Differential and Algebraic Geometry 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of PhysicsColumbia UniversityNew YorkU.S.A.
  2. 2.Institute for Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands

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