We present a detailed analysis of a scalar conformal four-point function obtained from AdS/CFT correspondence. We study the scalar exchange graphs on AdS d+1 and discuss their analytic properties. Using methods of conformal partial wave analysis, we present a general procedure to study conformal four-point functions in terms of exchanges of scalar and tensor fields. The logarithmic terms in the four-point function are connected to the anomalous dimensions of the exchanged fields. Comparison of the results from AdS d+1 graphs with the conformal partial wave analysis suggests a possible general form for the operator product expansion of scalar fields in the boundary CFT d . 1 1 The duality between string or M -theory compactifications on AdS d+1 and d-dimensional superconformal gauge theories suggested by AdS/CFT correspondence [1] has been the subject of intensive research over the past couple of years (for a recent review see [2]). Gradually, the emerging picture takes the form of the long-sought string/gauge theory relationship [3]. Recently, in a minkowsian version of the correspondence the d-dimensional conformal field theory (CFT) has been discussed in the context of local quantum field theory [4] defined on a standard (flat) compactified Minkowski space M c 1,d−1 . This space arises as the boundary of the AdS 1,d spacetime. The isometry group of both spaces is SO(d, 2) and the state space of the boundary CFT is related to the state space of the bulk theory [5].Such a view of the AdS/CFT correspondence implies that the known local structure of conformal field theory, (see for example [6] and references therein), is connected to the local structure of the the field (or string) theory living on AdS. In particular, harmonic analysis on the isometry group SO(d, 2) ("conformal partial wave analysis" CPWA), of n-point functions of the boundary CFT should be valid. This is equivalent to the existence of an operator product expansion (OPE) for the boundary CFT. Such expansions are convergent in a topology defined by the n-point functions on which they are applied (CPWA), or into which they are inserted (OPE). Perhaps the most well-known application ground for CPWA and OPEs is the (Euclidean) case d = 4, when the boundary CFT is the N = 4 SYM theory with gauge group SU (N ). In that case, the large-N , large-λ expansion (λ = g 2 Y M N with g Y M being the gauge coupling), corresponds to a perturbative form of the AdS theory in terms of the so-called "Witten graphs"[1]. Technical exploitations of the AdS/CFT correspondence are mainly based on this graphical expansion [8,9].Our aim in this work is to make a thorough investigation of a four-point function of scalar fields in the boundary CFT obtained from a graphical expansion in AdS. We choose to work in general dimensions d to ensure a broad applicability of our results. In Section 2 we set the stage for our study by considering a theory on AdS with a single cubic local interaction term.This may be viewed as the minimal AdS theory leading to a non-trivial four-point fu...
Operator product expansions are applied to dilaton-axion four-point functions. In the expansions of the bilocal fieldsΦΦ,CC andΦC, the conformal fields which are symmetric traceless tensors of rank l
We discuss the analytic properties of AdS scalar exchange graphs in the crossed channel. We show that the possible non-analytic terms drop out by virtue of non-trivial properties of generalized hypergeometric functions. The absence of non-analytic terms is a necessary condition for the existence of an operator product expansion for CFT amplitudes obtained from AdS/CFT correspondence.1
We develop a method of singularity analysis for conformal graphs which, in particular, is applicable to the holographic image of AdS supergravity theory. It can be used to determine the critical exponents for any such graph in a given channel. These exponents determine the towers of conformal blocks that are exchanged in this channel. We analyze the scalar AdS box graph and show that it has the same critical exponents as the corresponding CFT box graph. Thus pairs of external fields couple to the same exchanged conformal blocks in both theories. This is looked upon as a general structural argument supporting the Maldacena hypothesis.
We discuss the concept of composite fields in flat CFT as well as in the context of AdS/CFT. Furthermore we show how to represent Green functions using generalized hypergeometric functions and apply these techniques to four-point functions. Finally we prove an identity of U (1) Y symmetry for four-point functions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.