Bulk local states and crosscaps in holographic CFT

Yu Nakayama,Hirosi Ooguri

doi:10.1007/jhep10(2016)085

Abstract

In a weakly coupled gravity theory in the anti-de Sitter space, local states in the bulk are linear superpositions of Ishibashi states for a crosscap in the dual conformal field theory. The superposition structure can be constrained either by the microscopic causality in the bulk gravity or the bootstrap condition in the boundary conformal field theory. We show, contrary to some expectation, that these two conditions are not compatible to each other in the weak gravity regime. We also present an evidence to show that bulk local states in three dimensions are not organized by the Virasoro symmetry.

Highlights

The question we would like to address is how to take a linear superposition of Ishibashi states |φ over primary states |φ to construct a local state in the bulk AdS
We present an evidence to show that bulk local states in three dimensions are not organized by the Virasoro symmetry
A natural guess would be that it satisfies consistency conditions for a crosscap in CFT, in particular a bootstrap condition for crossing symmetry, which are analogous to the Cardy conditions on boundary states

Summary

Causality and cross-ratio

A crosscap state can be used to compute correlation functions of CFT on the real projective plane, which is usually considered in the Euclidean signature. The causality in AdS should be discussed in the Lorentzian signature. In order to compare the bootstrap condition on the projective plane and the microscopic causality in AdS, it is. Where coordinates on Sd−1 are denoted by Ω, which is identified with a unit vector in Rd. As a consequence of working in the global patch, the causal interpretation of the crosscap cross-ratio η, defined below, is slightly different from that discussed in [8,9,10,11,12] in the Poincare patch. Since the future light-cone from the center (t = 0, ρ = 0) of AdS reaches the boundary at t = π/2, a boundary point (t, Ω) is space-like separated from the center if and only if |t| < π/2. We can show that, when η > 1, at least one pair of the three points are space-like separated, modulo the 2π period in t

Microscopic causality in the bulk

Bootstrap condition on crosscaps

Gravity dual of crosscap states

Enhancement to the Virasoro symmetry

Virasoro enhancement at crosscaps

No Virasoro enhancement for bulk local states