Kinematic space and the orbit method

Abstract

Kinematic space has been defined as the space of codimension-$2$ spacelike extremal surfaces in anti de Sitter (AdS${}_{d+1}$) spacetime which, by the Ryu-Takayanagi proposal, compute the entanglement entropy of spheres in the boundary CFT${}_d$. 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 AdS${}_3$, 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.

arXiv comment25 pages, 5 figures

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• Complexity for Conformal Field Theories in General Dimensions
• Reparametrization modes, shadow operators, and quantum chaos in higher-dimensional CFTs
• Warped Schwarzian theory
• Berry phases, wormholes and factorization in AdS/CFT
• Virasoro Entanglement Berry Phases
• Holographic integral geometry with time dependence
• Bulk Locality and Asymptotic Causal Diamonds
• Berry curvature, horocycles and scattering states in $Ad{S}_3/CF{T}_2$
• Scanning space-time with patterns of entanglement