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

Abstract

We continue the study of four-point correlation functions by the hexagon tessellation approach initiated in [38] and [39]. We consider planar tree-level correlation functions in N = 4 supersymmetric Yang-Mills theory involving two non-protected operators. We find that, in order to reproduce the field theory result, it is necessary to include SU(N ) colour factors in the hexagon formalism; moreover, we find that the hexagon approach as it stands is naturally tailored to the single-trace part of correlation functions, and does not account for multi-trace admixtures. We discuss how to compute correlators involving double-trace operators, as well as more general 1/N effects; in particular we compute the whole next-to-leading order in the large-N expansion of tree-level BMN two-point functions by tessellating a torus with punctures. Finally, we turn to the issue of "wrapping", Lüscher-like corrections. We show that SU(N ) colour-dressing reproduces an earlier empirical rule for incorporating single-magnon wrapping, and we provide a direct interpretation of such wrapping processes in terms of N = 2 supersymmetric Feynman diagrams.

Octagons I: combinatorics and non-planar resummations

“…In matrix model theory, dualities between N → ∞ andÑ → 0 limits have in fact been found before, in the context of the theory of intersection numbers of moduli spaces of curves, see [26][27][28][29]. 12 As we will see below, the N → 0 limit will show up again and again in several simplified matrix model combinatorics in very amusing ways. It would be interesting to find a nice statistical mechanics application of this zero-color limit.…”

“…The basis of our computation is the (planar and non-planar) hexagonalization prescription for correlation functions [10][11][12][13][14][15]. The starting point of that prescription is a sum over all Wick contractions of the free gauge theory.…”

Non-planar data of $$ \mathcal{N} $$ = 4 SYM

“…Another motivation for computing non-planar structure constants is to further the understanding of non-planar integrability. Three-point functions can be computed at the planar level as a product of two integrable hexagon form-factors [37], and it was later understood that higher-point functions can also be decomposed into a weighted product of hexagon form-factors, both at the planar and non-planar level [38][39][40][41][42][43]. The integrability setup was tested at two loops for long operators and at one loop for short operators such as the 20 .…”

ABJM quantum spectral curve at twist 1: algorithmic perturbative solution

“…The most well understood theories at a moment are N = 4 SYM in four and N = 6 super Chern-Simons theory (ABJM model) in three dimensions [38]. The integrability based methods were also used in the study of quark-antiquark potential [39][40][41][42], expectation values of polygonal Wilson loops at strong coupling and beyond [43][44][45][46][47][48][49][50][51][52], eigenvalues of BFKL kernel [53][54][55][56], structure constants [57][58][59][60][61], correlation functions [57,[62][63][64][65][66][67][68][69][70][71], one-point functions of operators in the defect conformal field theory [72][73][74] and observables at finite temperature such as Hagedorn temperature of N = 4 SYM [75,76].…”