Holographic dual of the five-point conformal block
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Springer
Received: February 8, 2019 Accepted: April 23, 2019 Published: May 9, 2019
Sarthak Parikh Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, U.S.A.
E-mail: [email protected] 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. Keywords: AdS-CFT Correspondence, Conformal and W Symmetry ArXiv ePrint: 1901.01267
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2019)051
JHEP05(2019)051
Holographic dual of the five-point conformal block
Contents 1 Introduction
1 7 11 12
3 Discussion
16
A Propagator identities
18
1
Introduction
Conformal blocks are theory independent building blocks of conformal field theories which are fixed entirely from conformal invariance. Equipped with the CFT data (i.e. the spectrum and the OPE coefficients of any theory), the knowledge of conformal blocks permits the construction of all correlators in the theory. Given their fundamental importance in CFTs, it is important to understand and investigate the holographic duals of conformal blocks in the context of AdS/CFT. Such a task was recently undertaken for four-point global conformal blocks in any dimension with external scalar operators [1] (with further generalizations in refs. [2–13]) and for Virasoro blocks in AdS3 /CFT2 [14–24]. Nevertheless, the holographic interpretation of higher-point global conformal blocks has remained unresolved so far, and fraught with various technical obstructions. With the hope that new tools may provide new clues, in this paper, we appeal to the framework of p-adic AdS/CFT [6, 25, 26] which offers a significantly less complicated setting for tackling the same problem. Indeed, it turns out, the problem of various higher-point blocks is relatively easily addressed over the p-adics [27], and owing to the close ties with the usual AdS/CFT setup (see, e.g., refs. [6, 25, 26, 28–35]) we are able to extract key new insights into the analogous calculation over reals. The p-adic results generalize the result of ref. [6], and are discussed in ref. [27]. The p-adic five-point case, related to the subject of this paper, is briefly summarized later in this section. In this paper, inspired by the p-adic result, we establish the holographic dual of the global five-point block with external scalar operators in the usual (real) AdS n+1 /CFTn setup, generalizing the result of ref.
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