The octagon as a determinant

Abstract

The computation of a certain class of four-point functions of heavily charged BPS operators boils down to the computation of a special form factor -the octagon. In this paper, which is an extended version of the short note [1], we derive a non-perturbative formula for the square of the octagon as the determinant of a semi-infinite skew-symmetric matrix. We show that perturbatively in the weak coupling limit the octagon is given by a determinant constructed from the polylogarithms evaluating ladder Feynman graphs. We also give a simple operator representation of the octagon in terms of a vacuum expectation value of massless free bosons or fermions living in the rapidity plane. arXiv:1905.11467v1 [hep-th] 27 May 20191 The fact that the sum over virtual particles can be written as a Fredholm pfaffian has been noticed before in [14].(1.7) 2 The integrability of the fishnet Feynman graphs has been first established by A. Zamolodchikov [21]. 3 By Zhukovsky plane we mean the rapidity plane with two simple branch points at u = ±2g. A similar operator representation has been proposed in [22] for the three-point function of non-BPS operators.

Origin of the Six-Gluon Amplitude in Planar N=4 Supersymmetric Yang-Mills Theory

“…Note that Γ oct has an expansion in powers of π 2 only (through 7 loops at least). Furthermore, it agrees with the exact [31] anomalous dimension controlling the light-like limit of the octagon [27][28][29][30],…”

“…From the growth rate of their perturbative coefficients, all these quantities appear to have same radius of convergence, g 2 c = 1/16, as Γ cusp [26]. The point α = 0 corresponds to the octagon [27][28][29][30].…”

“…Different tilt angles generate different anomalous dimensions controlling logarithmicallyenhanced terms in the amplitude. Intriguingly, one of the anomalous dimensions also appears in the light-like limit of the octagon [27][28][29][30][31][32], a correlation function of four operators with large R charge. We also predict the nonlogarithmic term, as well as the coefficient ρ controlling a "cosmic" amplitude normalization [33].…”

Three-point functions at strong coupling in the BMN limit

“…It would also be interesting to compare our findings with the formulae obtained in [30,31] for large-charge 4pt functions. Although both arise from hexagons the comparison is not immediate since they run with different transfer matrices, which is also why they describe different observables of the boundary theory.…”

“…For more magnons, it proves convenient to use the Pfaffian formula for the hexagons [53], see also [30,31] for recent discussions. Namely, defining…”