The gravity dual of Lorentzian OPE blocks

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.

Regge OPE blocks and light-ray operators

“…Meanwhile, we will be focused on the vacuum OPE blocks and their implications on a non-vacuum state, leaving the determination of the non-vacuum part for future studies. We employ a new representation of the vacuum OPE block which has a geometric interpretation as an AdS propagating field smeared over the geodesic between the boundary points x 1 and x 2 in an AdS spacetime, initially obtained for a scalar channel in [15][16][17] and generalized to any channel recently in [18]. We use the new representation inside four-point functions to see if it leads to the known behaviors of conformal blocks.…”

“…Our derivation closely follows the relevant works [19,20] where similar results were obtained in a slightly different way. These works started with a pair of timelike-separated operators, took the Regge-like limit of the timelike OPE block B ♦ ∆,J proposed by [15,16], which differs from the spacelike OPE block B ∆,J of [18] we use in this paper, and then analytically continued the result to the spacelike configuration. To fill the gap between the two approaches, in section 4, we compare B ♦ ∆,J with another form of the timelike OPE block B T ∆,J obtained by analytically continuing the spacelike one B ∆,J .…”

“…The Lorentzian OPE block was derived a long time ago by [5,7,9,36], and has attracted renewed interests in connection with its holographic description on the AdS d+1 spacetime in literature [15][16][17] where the scalar block (J = 0) has been studied extensively. A more complete analysis including higher spin cases has been undertaken in a recent paper [18], where the OPE block is shown to take different forms depending on the causal relation of the two scalar primaries for which the OPE is taken.…”

“…One can proceed with this representation and rewrite the three-point function by introducing a Feynman parametrization ξ. By exchanging the order of integration between p and ξ one ends up with an integral representation of the spacelike OPE block [18]:…”

“…We do not bother to write the complete expression of Φ ∆,J as it is unnecessary in the following discussion. The interested reader is referred to [18] for the detail. In momentum space the l th term takes up to a constant the form:…”