Abstract
We present the holographic object which computes the five-point global conformal block in arbitrary dimensions for external and exchanged scalar operators. This object is interpreted as a weighted sum over infinitely many five-point geodesic bulk diagrams. These five-point geodesic bulk diagrams provide a generalization of their previously studied four-point counterparts. We prove our claim by showing that the aforementioned sum over geodesic bulk diagrams is the appropriate eigenfunction of the conformal Casimir operator with the right boundary conditions. This result rests on crucial inspiration from a much simpler p-adic version of the problem set up on the Bruhat-Tits tree.
Highlights
We present the holographic object which computes the five-point global conformal block in arbitrary dimensions for external and exchanged scalar operators
We have proposed the holographic object which computes the global fivepoint conformal block in any spacetime dimension
Explicit expressions for the global five-point conformal block are already known in the literature, obtained via the monodromy method [20] and the shadow formalism [42], so it is natural to ask how our results line up against previous results
Summary
Where P k stands for k applications of the momentum generators Pμ to obtain the descendants of O. where P k stands for k applications of the momentum generators Pμ to obtain the descendants of O The upshot of this exercise is that the following geodesic bulk integral, W∆∆1,...,∆4 (xi) ≡. W ∈γ34 where G, Kare bulk-to-bulk and bulk-to-boundary propagators respectively, computes the conformal partial wave associated with the exchange of an operator of dimension ∆ between external insertions O1, O2 and O3, O4 [1] (see figure 2). Where the bulk point w = w(λ) is parametrized by λ running along the boundary anchored geodesic γij between boundary points xi and xj. To obtain the holographic dual of the five-point conformal partial wave defined via the projection, W∆∆,1∆,...,∆5 (xi)
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