Mellin Amplitudes for Supergravity onAdS5×S5

Abstract

We revisit the calculation of holographic correlation functions in IIB supergravity on AdS5 × S 5 . Results for four-point functions simplify drastically when expressed in Mellin space. We conjecture a compact formula for the four-point functions of one-half BPS single-trace operators of arbitrary weight. Our methods rely on general consistency conditions and eschew detailed knowledge of the supergravity effective action.PACS numbers: 11.25. Tq, 11.15.Pg, 11.55.Bq Introduction. Despite almost two decades of relentless efforts, we are still far from harnessing the full computational power of the AdS/CFT correspondence. In the canonical duality [1][2][3] between N = 4 super Yang-Mills (SYM) theory and IIB string theory on AdS 5 × S 5 , the bulk description is most tractable in the classical supergravity regime, which describes planar SYM theory at large 't Hooft coupling. Supergravity is however still a complicated non-linear theory, and only the simplest observables have been computed so far. In this letter we revisit the holographic calculation of four-point correlation functions of one-half BPS single-trace operators [4]. In the supergravity limit, there is a straightforward algorithm that computes them as a sum of tree-level Witten diagrams, whose vertices are encoded in the AdS 5 effective action [5] obtained by Kaluza-Klein (KK) reduction of IIB supergravity on S 5 .

Towards 4-point correlation functions of any 1 2 $$ \frac{1}{2} $$ -BPS operators from supergravity

“…Analysing the behaviour of exchange graphs for large values of Mandelstam variables one concludes that they grow linearly, hence the conjecture by [4]. As to the contact graphs, their linear asymptotic growth is guaranteed provided the quartic Lagrangian vertices contain fields s I with a number of derivatives not higher than two.…”

“…On the one hand, this gives evidence in favour of the conjecture [4] and, on the other hand, provides a first step towards actual evaluation of the four-point correlation function of arbitrary weight α , where α is the tangent index of the five-sphere. By unravelling some intricate identity which involves a sum of integrals over vector harmonics, we managed to rewrite the contribution of the latter via similar sums but involving scalar harmonics only.…”

“…Recently the authors of [4] conjectured an interesting formula for the Mellin representation of the four-point correlation function of arbitrary weight 1 2 -BPS operators in the supergravity limit. In addition to the expected OPE behavior in this limit and to the agreement with an earlier proposal [5] on the structure of the four-point correlation functions of equal weight Leaving technical complications out of the discussion, to derive the corresponding correlation function and in this way to prove the conjecture by [4], one has to use the quartic effective action for Kaluza-Klein modes of type IIB supergravity compactified on the fivesphere S 5 [6]. We recall that Here A exchange comprises the sum of all exchange graphs which are generated by cubic Lagrangian vertices involving two scalars s I and any other supergravity field which is…”

Loop corrections for Kaluza-Klein AdS amplitudes

“…While there has been much study of treelevel bulk processes [4][5][6][7][8][9][10][11][12], until recently there had not been much work on loop corrections (see [13,14] for some recent work in this direction). Essentially this is because such loop computations are extremely challenging from the bulk approach.…”

Colour-dressed hexagon tessellations for correlation functions and non-planar corrections

“…More recently, it was understood that fourpoint correlators capture information on locality in the bulk [32,33], which makes their investigation particularly interesting. Indeed in recent times a number of new results have appeared in this field [34][35][36]. In principle, four-point functions are fixed via the operator product expansion (OPE) in terms of the lower-point correlators.…”