Journal of High Energy Physics

, 2019:268 | Cite as

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

  • Christian Baadsgaard Jepsen
  • Sarthak ParikhEmail author
Open Access
Regular Article - Theoretical Physics


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 higher-point 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 direct-channel 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.


AdS-CFT Correspondence 1/N Expansion Conformal and W Symmetry Conformal Field Theory 


Open Access

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© The Author(s) 2019

Authors and Affiliations

  1. 1.Joseph Henry LaboratoriesPrinceton UniversityPrincetonU.S.A.
  2. 2.Division of Physics, Mathematics and AstronomyCalifornia Institute of TechnologyPasadenaU.S.A.

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