Witten diagrams revisited: the AdS geometry of conformal blocks
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Witten diagrams revisited: the AdS geometry of
Eliot Hijano 0 1 3
Per Kraus 0 1 3
Eric Perlmutter 0 1 2
River Snively 0 1 3
0 Princeton , NJ 08544 , U.S.A
1 Los Angeles , CA 90095 , U.S.A
2 Department of Physics, Princeton University
3 Department of Physics and Astronomy, University of California
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.
AdS-CFT Correspondence; Conformal and W Symmetry
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1 Introduction 2
Conformal blocks, holographic CFTs and Witten diagrams
CFT four-point functions and holography
A Witten diagrams primer 2.2.1 2.2.2
Mellin space
Looking ahead
3
The holographic dual of a scalar conformal block
3.3
Comments
3.2.1
3.2.2
3.2.3
3.3.1
3.3.2
3.3.3
Proof by direct computation Proof by conformal Casimir equation The Casimir equation Embedding space
Geodesic Witten diagrams satisfy the Casimir equation
Geodesic versus ordinary Witten diagrams
Simpli cation of propagators and blocks
Relation to Mellin space
2.1
2.2
2.3
2.4
3.1
3.2
4.1
4.2
4.3
4.4
5.1
5.2
5.3
5.4
5.5
5.6
5.7
Known results
Geodesic Witten diagrams with spin-` exchange: generalities
Evaluation of geodesic Witten diagram: spin-1
Evaluation of geodesic Witten diagram: spin-2
General `: proof via conformal Casimir equation
Comparison to double integral expression of Ferrara et al.
Decomposition of spin-1 Witten diagram into conformal blocks
5.7.1
5.7.2
AAA
AA and A A
{ i {
Logarithmic singularities and anomalous dimensions
What has been computed?
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The conformal block decomposition of scalar Witten diagrams
An AdS propagator identity
Four-point contact diagram
Four-point exchange diagram
Further analysis
4.4.1
4.4.2
OPE factorization
Recovering logarithmic singularities
4.5
Taking stock
5
Spinning exchanges and conformal blocks
5.7.3
5.7.4
Summary
6
Discussion and future work
1
Introduction
The conformal block decomposition of correlation functions in conformal eld theory is a
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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{7]. The latter is expressed as a list of primary operators and the OPE coe cients
amongst them. The use of conformal blocks in the study of CFT correlation functions
therefore eliminates redundancy, as heavily utilized, for instance, in recent progress made
in the conformal bootstrap program, e.g. [8, 9].
In the AdS/CFT correspondence [10{12], the role of conformal blocks has been
somewhat neglected. 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{20], and has been re ned in recent years through the
introduction of Mellin space technology [21{27]. In examining this body of work, however,
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 e cient implementation of the dual conformal block decomposition, as it
would remove the necessity of actually computing the full Witten diagram explicitly. The
absence of such a simpler method would indicate a surprising failure of our
understanding of AdS/CFT: after all, conformal blocks are determined by conformal symmetry, the
matching of which is literally the most basic element in the holographic dictionary.
In this paper we present an appealingly simple answer to the above question, and
demonstrate its utility via streamlined computations of Witten diagrams. More precisely,
we will answer this question in the case of four-point correlation functions of scalar
operators, but we expect a similar story to hold in general. The answer is that conformal blocks
are computed by \geodesic Witten diagrams". 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 bounda (...truncated)