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

Abstract

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.

Unitarity methods in AdS/CFT

“…These results are also consistent with those of [72,73] which used geodesic Witten diagrams [76]. Now we turn to the crossed channel decomposition.…”

“…There have been a few studies of higher-point correlators, e.g. [71][72][73]77] but they have received much less focus than four-point functions. However, as we study higher-loop, fourpoint correlators in AdS/CFT they inevitably appear when cutting certain Witten diagrams.…”

“…27 For example, the leading log comes from a term (γ (1) n, ) 3 in dDisc(G 2−loop ). 28 For further work on higher-point tree diagrams and conformal blocks, see [35,[71][72][73][74][75]…”

A multipoint conformal block chain in d dimensions

“…[54]). Geodesic diagram representations have also been obtained for the same five-point block in general spacetime dimensions [55], as well as for the d-dimensional six-point scalar conformal block involving scalar exchanges in a different channel called the OPE channel [56].…”

Towards the higher point holographic momentum space amplitudes. Part II. Gravitons

“…3 The study of the scattering amplitudes of gauge theories and gravity has revealed remarkably simple structures in flat space and has led to menagerie of basis such as twistors and geometric formulations like the amplituhedron [15][16][17][18][19]. Additionally, in the last several years we have witnessed inspiring representations which have helped in showing similarities between scattering amplitudes in flat space and their AdS counterparts [9,[20][21][22][23][24][25][26][27][28][29][30][31][32][33]. One related insight has come out of the recent investigation of holographic momentum space which indicates an interesting connection between momentum space Witten diagrams and flat space scattering amplitudes.…”