Aspects of the conformal operator product expansion in $\mathrm{AdS}$/$\mathrm{CFT}$ correspondence

Abstract

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...

Near-extremal correlators and vanishing supergravity couplings in AdS/CFT

“…where 11) and the square roots account for the large N Wick factors and the normalization. The prime in the four point function in (3.10) indicates that a factorized free-field contribution to this four point function has been explicitly written in the second term of the equation and must not be included again.…”

“…For near-extremality, m ≥ 2, the situation is more complex as we now exemplify for the case m = 2. Here the E 2 4 correlator O 2 O 2 O 2 O 2 is known to be renormalized to order g 2 in field theory [7,8] (for a recent calculation to three loops see [9]) and in the large N strong coupling limit in supergravity [10,11,12]. Indeed E 2 4 correlators have no special factored free-field limits, which suggests that one must examine E 2 5 functions to find non-renormalization.…”

Operator product expansion of the lowest weight CPOs in =4 at strong coupling

“…Computing 4-point functions [13]- [26] in the supergravity approximation in general requires the derivation of the supergravity action up to fourth order. The part of the action relevant to the massless modes, corresponding to the dilaton and axion fields, was already known as pointed out in [13] where the calculation of the corresponding 4-point functions was initiated.…”

AdS box graphs, unitarity and operator product expansions

“…there are terms from cutting external lines in AdS d+1 . As was argued [13] in the shadow term in CFT d and the external line terms in AdS d+1 are necessary to guarantee analytic behavior in the crossed channel.…”

“…Partly with techniques developed first for CFT in flat space, the exchange graph was calculated and studied in a series of works [12,13]. The results of all such calculations were finally expressed in terms of generalized hypergeometric functions.…”