Anatomy of geodesic Witten diagrams

Self Cite

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.

Section: Contentssupporting

confidence: 74%

The gravity dual of Lorentzian OPE blocks

Chen

Kobayashi

et al. 2020

Self Cite

“…The authors use different techniques to address similar questions. Shorlty following the appearance of our work, two papers containing some overlap with our own appeared [61,62].…”

Section: Jhep11(2017)060mentioning

confidence: 96%

Spinning geodesic Witten diagrams

Dyer¹,

Freedman²,

Sully³

2017

We present an expression for the four-point conformal blocks of symmetric traceless operators of arbitrary spin as an integral over a pair of geodesics in Anti-de Sitter space, generalizing the geodesic Witten diagram formalism of Hijano et al.[1] to arbitrary spin. As an intermediate step in the derivation, we identify a convenient basis of bulk threepoint interaction vertices which give rise to all possible boundary three point structures. We highlight a direct connection between the representation of the conformal block as geodesic Witten diagram and the shadow operator formalism.

“…[62,94,93,97,98,99,96,101,102]). In arbitrary spacetime dimensions, it is also natural to consider the generalizations of the holographic duals of the higher-point scalar blocks of this paper to those involving external and exchanged spinning operators, along the lines of the four-point case [83,84,85,86,88,89,90]. Progress along this direction may also aid the technically challenging task of the holographic reconstruction of the classical bulk action for higher spin gravity theories beyond quartic interaction vertices [128,129,130].…”

Section: Spectral Decomposition Of Ads Diagramsmentioning

confidence: 99%

“…5 The four-point geodesic diagram is a four-point exchange AdS diagram, except with both AdS integrations replaced by geodesic integrals over boundary anchored geodesics joining pairs of boundary insertion points. Such a holographic object computes precisely the four-point global conformal block [14,83,84,85,86,72,87,88,89,90] (see also Refs. [91,92] for an alternate point of view).…”

Section: Introductionmentioning

confidence: 99%

Propagator identities, holographic conformal blocks, and higher-point AdS diagrams

Jepsen

Parikh

2019

Conformal blocks are the fundamental, theory-independent building blocks in any CFT, so it is important to understand their holographic representation in the context of AdS/CFT. We describe how to systematically extract the holographic objects which compute higherpoint global (scalar) conformal blocks in arbitrary spacetime dimensions, extending the result for the four-point block, known in the literature as a geodesic Witten diagram, to five-and six-point blocks. The main new tools which allow us to obtain such representations are various higher-point propagator identities, which can be interpreted as generalizations of the well-known flat space star-triangle identity, and which compute integrals over products of three bulk-to-bulk and/or bulk-to-boundary propagators in negatively curved spacetime. Using the holographic representation of the higher-point conformal blocks and higher-point propagator identities, we develop geodesic diagram techniques to obtain the explicit directchannel conformal block decomposition of a broad class of higher-point AdS diagrams in a scalar effective bulk theory, with closed-form expressions for the decomposition coefficients. These methods require only certain elementary manipulations and no bulk integration, and furthermore provide quite trivially a simple algebraic origin of the logarithmic singularities of higher-point tree-level AdS diagrams. We also provide a more compact repackaging in terms of the spectral decomposition of the same diagrams, as well as an independent discussion on the closely related but computationally simpler framework over p-adics which admits comparable statements for all previously mentioned results.