Abstract
We develop a new method for decomposing Witten diagrams into conformal blocks. The steps involved are elementary, requiring no explicit integration, and operate directly in position space. Central to this construction is an appealingly simple answer to the question: what object in AdS computes a conformal block? The answer is a "geodesic Witten diagram," which is essentially an ordinary exchange Witten diagram, except that the cubic vertices are not integrated over all of AdS, but only over bulk geodesics connecting the boundary operators. In particular, we consider the case of four-point functions of scalar operators, and show how to easily reproduce existing results for the relevant conformal blocks in arbitrary dimension.
Highlights
The conformal block decomposition of correlation functions in conformal field theory is a powerful way of disentangling the universal information dictated by conformal symmetry from the “dynamical” information that depends on the particular theory under study; see e.g. [1,2,3,4,5,6,7]
The extraction of spectral and OPE data of the dual CFT from a holographic correlation function, as computed by Witten diagrams [12], was addressed early on in the development of the subject [13,14,15,16,17,18,19,20], and has been refined in recent years through the introduction of Mellin space technology [21,22,23,24,25,26,27]. In examining this body of work, one sees that a systematic method of decomposing Witten diagrams into conformal blocks is missing
A rather natural question appears to have gone unanswered: namely, what object in AdS computes a conformal block? A geometric bulk description of a conformal block would greatly aid in the comparison of correlators between AdS and CFT, and presumably allow for a more efficient implementation of the dual conformal block decomposition, as it would remove the necessity of computing the full Witten diagram explicitly
Summary
The conformal block decomposition of correlation functions in conformal field theory is a powerful way of disentangling the universal information dictated by conformal symmetry from the “dynamical” information that depends on the particular theory under study; see e.g. [1,2,3,4,5,6,7]. The main feature of a geodesic Witten diagram that distinguishes it from a standard exchange Witten diagram is that in the former, the bulk vertices are not integrated over all of AdS, but only over geodesics connecting points on the boundary hosting the external operators This representation of conformal blocks in terms of geodesic Witten diagrams is valid in all spacetime dimensions, and holds for all conformal blocks that arise in four-point functions of scalar operators belonging to arbitrary CFTs (and probably more generally). The most efficient way to prove that geodesic Witten diagrams compute conformal partial waves is to establish that they are the correct eigenfunctions This turns out to be quite easy using embedding space techniques, as we will discuss. ; the reader is referred to [35, 36] and references therein for foundational material
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