How to succeed at Witten diagram recursions without really trying

Abstract

Witten diagrams are basic objects for studying dynamics in AdS space, and also play key roles in the analytic functional bootstrap. However, these diagrams are notoriously hard to evaluate, making it extremely difficult to search for recursion relations among them. In this note, we present simple methods to obtain recursion relations for exchange Witten diagrams from conformal block recursion relations. We discover a variety of new relations, including the dimensional reduction formulae for exchange Witten diagrams. In particular, we find a five-term recursion relation relating exchange Witten diagrams in d and d − 2 dimensions. This gives the holographic analogue of a similar formula for conformal blocks due to Parisi-Sourlas supersymmetry. We also extend the analysis to two-point functions in CFTs with conformal boundaries, and obtain similar results.

More on holographic correlators: twisted and dimensionally reduced structures

“…This turned out to be related to the linear combination one needs in order to repackage bosonic conformal blocks into superblocks of the Parisi-Sourlas superconformal algebra. As shown in [15,19], correlation functions in d dimensions with this supersymmetric property can be interpreted as those of a theory in d − 2 dimensions. For the theories studied here, we were able to use this relation twice so that the…”

“…Nevertheless, we find strong evidence indicating that this reduction phenomenon is intimately related to the Parisi-Sourlas dimensional reduction, which relates a d dimensional theory with Parisi-Sourlas supersymmetry and a d − 2 dimensional non-supersymmetric theory. The connection comes from a dimensional reduction formula for exchange Witten diagrams in AdS d+1 and AdS d−1 , which was shown in [19] as the consequence of the holographically realized Parisi-Sourlas supersymmetry. It turns out that using the dimensional reduction formula twice gives precisely (1.2).…”

“…can be interpreted as a single superconformal block under Parisi-Sourlas supersymmetry. Later it was pointed out in [19] that the Parisi-Sourlas supersymmetry can also be realized holographically. This gives rise to similar dimensional reduction identities for exchange Witten diagrams, obtained by replacing conformal blocks with the corresponding exchange…”

“…While the sum over I h,n (α) looks complicated a priori, with P n (α) being in principle arbitrary polynomials of degree n, 19 for the specific values of n that enter the sum (5.13) they turn out to have a particularly simple form, in which the dependence on α is almost completely factorized:…”

“…19) in terms of the factors from(3.10). By scrutinizing the gamma functions in (3.25), we can see that the inner sum only goes up to E − k 2 +k 3 −p 2 instead of E. This allows us to perform the reflection j…”