The gravity dual of Lorentzian OPE blocks

TL;DR

This paper explores the gravitational dual of Lorentzian OPE blocks in conformal field theories, revealing their geometric representation in Lorentzian AdS space and a novel hyperboloid structure for time-like separations.

Contribution

It introduces a new geometric interpretation of Lorentzian OPE blocks, including a hyperboloid dual space for time-like separated operators, extending the holographic understanding of CFTs.

Findings

OPE blocks are integrals of higher spin fields along geodesics for space-like separations.

For time-like separations, OPE blocks are represented on a hyperboloid with an extra time coordinate.

In 2D, the hyperboloid duality reproduces known results via a kinematical duality.

Abstract

We consider the operator product expansion (OPE) structure of scalar primary operators in a generic Lorentzian CFT and its dual description in a gravitational theory with one extra dimension. The OPE can be decomposed into certain bi-local operators transforming as the irreducible representations under conformal group, called the OPE blocks. We show the OPE block is given by integrating a higher spin field along a geodesic in the Lorentzian AdS space-time when the two operators are space-like separated. When the two operators are time-like separated however, we find the OPE block has a peculiar representation where the dual gravitational theory is not defined on the AdS space-time but on a hyperboloid with an additional time coordinate and Minkowski space-time on its boundary. This differs from the surface Witten diagram proposal for the time-like OPE block, but in two dimensions we reproduce it consistently using a kinematical duality between a pair of time-like separated points and space-like ones.