Poking Holes in AdS/CFT: Bulk Fields from Boundary States

Abstract

We propose an intrinsic CFT definition of local bulk operators in AdS3/CFT2 in terms of twisted Ishibashi boundary states. The bulk field Phi(X) creates a cross cap, a circular hole with opposite edge points identified, in the CFT space-time. The size of the hole is parameterized by the holographic radial coordinate y. Our definition is state-independent, non-perturbative, and does not presume or utilize a semi-classical bulk geometry. We argue that, at large central charge, the matrix element between highly excited states satisfies the bulk wave equation in the AdS black hole background.

Figures (2)

• Figure 1: In a semi-classical treatment, the matrix element (\ref{['matrixelt']}) equals the 2D Liouville action associated to a hyperpoblic cylinder with a single cross-cap. The moduli space of this hyperbolic surface is isomorphic to the BTZ black hole space-time.
• Figure 2: The CFT amplitude with the insertion of an Ishibashi boundary state (left) and its Schottky double (right). Due to the projection onto the conformal sector $h$ in the intermediate channel, the amplitude is given by a single conformal block.