Bootstrapping correlation functions in N = 4 $$ \mathcal{N}=4 $$ SYM

Abstract

We describe a new approach to computing the chiral part of correlation functions of stress-tensor supermultiplets in N = 4 SYM that relies on symmetries, analytic properties and the structure of the OPE only. We demonstrate that the correlation functions are given by a linear combination of chiral N = 4 superconformal invariants accompanied by coefficient functions depending on the space-time coordinates only. We present the explicit construction of these invariants and show that the six-point correlation function is fixed in the Born approximation up to four constant coefficients by its symmetries. In addition, the known asymptotic structure of the correlation function in the light-like limit fixes unambiguously these coefficients up to an overall normalization. We demonstrate that the same approach can be applied to obtain a representation for the six-point NMHV amplitude that is free from any auxiliary gauge fixing parameters, does not involve spurious poles and manifests half of the dual superconformal symmetry.

The correlahedron

“…This form was directly lifted from the corresponding formula found in [42] in analytic superspace. It is clear from this example that the construction of superconformal invariants on analytic superspace has a direct uplift into the correlahedron space more generally.…”

“…Many correlators have by now been constructed explicitly writing down forms with the correct singularity structure and showing that they satisfy appropriate consistency properties. In particular, for maximal k = n − 4, they have been constructed up to n = 14 (equivalent to 10 loop four-point correlators), and the next to maximal case, k = 1, n = 6 has also been constructed [8,[33][34][35][36][37][38][39][40][41][42]. We will see here that given these expressions, there is a simple procedure to uplift them directly (and uniquely) into correlahedron volume forms on Gr(n+k, 4+n+k).…”

“…The only non-maximally nilpotent correlator currently known explicitly is the six point k = 1 correlator. This was derived in analytic superspace in [42] and also has a straightforward lift to a correlahedron volume form. This correlahedron lives in the space Y ∈ Gr(7, 11) so all the angle brackets are 11-brackets in the following expression.…”

“…The correlator G 6;1 was given in [42] in terms of nilpotent superconformal invariants I ijkl;αβγδ in analytic superspace. Lifting the correlator to the 11-dimensional correlahedron space, these nilpotent invariants become the following product of 4 11-brackets:…”

“…One interesting consequence of this correlahedron formulation of the correlator as an object in 11-dimensions is that it manifests highly non-trivial identities involving these invariants. It was observed in [42] that the invariants satisfy the non-trivial identity 6 i=1 X iα I ijkl;αβγδ = 0 (for all j, k, l, M, β, γ, δ) ,…”