Holographic four-point functions in the (2, 0) theory
Accepted: June
Holographic four-point functions in the (2; 0) theory
Leonardo Rastelli 0 1
Xinan Zhou 0 1
Theory, Scattering Amplitudes
0 Stony Brook , 11794, NY , U.S.A
1 C.N. Yang Institute for Theoretical Physics, Stony Brook University
We revisit the calculation of holographic correlators for eleven-dimensional S4. Our methods rely entirely on symmetry and eschew detailed knowledge of the supergravity e ective action. By an extension of the position space approach developed in [1, 2] for the AdS5 S5 background, we compute four-point correlators of one-half BPS operators for identical weights k = 2; 3; 4. The k = 2 case corresponds to the four-point function of the stress-tensor multiplet, which was already known, while the other two cases are new. We also translate the problem in Mellin space, where the solution of the superconformal Ward identity takes a surprisingly simple form. We formulate an algebraic problem, whose (conjecturally unique) solution corresponds to the general one-half BPS four-point function.
AdS-CFT Correspondence; Conformal and W Symmetry; Conformal Field
-
supergravity on AdS7
1 Introduction 2 3 4
Mellin space
3.1
3.2
3.3
3.4
4.1
4.2
4.3
Superconformal kinematics of four-point functions
Position space
The position space method The k = 2 four-point function The k = 3 four-point function The k = 4 four-point function
Consistency conditions and the algebraic problem
Solutions from the position space method
A R-symmetry polynomials
B Fixing the overall constant
C Four-point functions of the Wn!1 algebra
servables to be computed using the AdS/CFT dictionary, but only recently [1, 2] e cient
calculational methods have begun to be developed. The traditional recipe is based on a
perturbative expansion in Witten diagrams, which becomes very cumbersome (already at tree
level) for n-point correlators with n > 4. Prior to our work, only a few four-point correlators
of Kaluza-Klein (KK) modes were known, to wit (focussing for de niteness on the
maximally supersymmetric backgrounds): a handful of cases in the AdS5
S5 background [3{9];
just the four-point function of the lowest KK mode (the stress-tensor multiplet) in the
AdS7
S4 background [10]; and no results whatsoever in the AdS4
S7 background.
The traditional method has two sources of computational complexity: the need for
explicit expressions of the vertices in the supergravity e ective action; and the proliferation
of exchange Witten diagrams as the KK level is increased. In [1, 2] we introduced new
calculational tools to circumvent these di culties, for the case of AdS5
A rst approach, which we refer to as the \position space method", leverages the special
feature of the AdS5
S5 background that all exchange Witten diagrams can be written as
nite sums of contact diagrams. One can then write an ansatz for the four-point correlator
S5 supergravity.
{ 1 {
as a sum of contact diagrams, and determine their relative coe cients by imposing
superconformal symmetry, with no need for a detailed knowledge of the e ective supergravity
action. While simpler than the standard perturbative recipe, the position space method
also quickly runs out of steam as the KK level is increased. What's worse, the answer
takes a completely unintuitive form, with no simpe general pattern. The second, more
powerful approach of [1, 2] uses the Mellin representation of conformal correlators [11, 12].
Tree-level holographic correlators in AdS5
S5 are rational functions of Mandelstam-like
invariants, with poles and residues controlled by OPE factorization, in close analogy with
tree-level at space scattering amplitudes. Superconformal symmetry is made manifest
by solving the superconformal Ward identity in terms of an \auxiliary" Mellin amplitude.
The consistency conditions that this amplitude must satisfy de ne a very constrained
algebraic problem, which very plausibly admits a unique solution. While the position space
method is implemented on a case-by-case basis for di erent correlators, the Mellin
algebraic problem takes a universal form. We were able to solve the problem in one fell swoop
for all half-BPS four-point function with arbitrary weights | a feat extremely di cult to
replicate in position space.
The goal of this note is to extend these techniques to eleven-dimensional supergravity
on AdS7
S4. This is a background of extraordinary physical interest, as it provides a
dual description of the mysterious six-dimensional (2; 0) theory at large n, and a prime
target for our methods, since we expect maximal supersymmetry to constrain the
treelevel holographic correlators uniquely. At a more technical level, this background enjoys
the same \truncation conditions" as AdS5
S5, such that exchange Witten diagrams can
be written as nite sums of contact diagrams | equivalently, tree-level Mellin amplitudes
are rational functions of the Mandelstam invariants. By contrast, the truncation conditions
do not hold for the AdS4
S7 background, and new too (...truncated)