Bootstrapping the Simplest Correlator in Planar N=4 Supersymmetric Yang-Mills Theory to All Loops

Abstract

We present the full form of a four-point correlation function of large BPS operators in planar N = 4 Super Yang-Mills to any loop order. We do this by following a bootstrap philosophy based on three simple axioms pertaining to (i) the space of functions arising at each loop order, (ii) the behaviour in the OPE in a double-trace dominated channel and (iii) the behaviour under a double null limit. We discuss how these bootstrap axioms are in turn strongly motivated by empirical observations up to nine loops unveiled through integrability methods in our previous work [9] on this simplest correlation function.

The octagon as a determinant

“…We claim that to any loop order the pfaffian can be recast as a determinant of another matrix. From the determinant representations one confirms by direct computation the conjecture of [18] that the perturbative expansion of the octagon can be recast as a multilinear combination of ladder integrals…”

“…The matrix elements of R simplify in this limit and for the determinant we obtain, after massive cancellations, the asymptotics (1.36). We checked that the perturbative expansion ofΓ(g) is in agreement with [18]. To computeΓ(g) to N loops it is sufficient to keep the first N terms in the series (1.37) and truncate R to an N × N matrix.…”

“…In [18] it was conjectured that the logarithm of the octagon enjoys the following simple asymptotics,…”

“…The form (1.2) of the perturbative octagon carries some resemblance to the result of Basso and Dixon [19] obtained for the fishnet limit of the N = 4 SYM [20] 2 . The analytic expression obtained in [19] for the fishnet has the form of a single determinant of ladders, while bootstraping the Ansatz (1.2) one obtains [18] a series in the minors of the semi-infinite matrix…”

One-loop amplitudes in AdS5×S5 supergravity from $$ \mathcal{N} $$ = 4 SYM at strong coupling

“…Future directions include a more detailed investigation of the Mellin space representation of our one-loop functions, which would extend the analysis of 2222 in [11], as well as the possibility of pushing our bootstrap program to two loops. It would be fascinating if the results we obtain in the large N expansion could be compared (possibly taking into account also the α ′ corrections) to the results based on integrable methods [50][51][52][53][54]. We also emphasise that our fresh new look at free N = 4 SYM, especially our understanding of single particle operators and generalised tree-level functions, suggests a different way to approach the mysterious six-dimensional (2, 0) theory, which has been recently studied from a holographic perspective in several papers [55][56][57][58].…”