Crossing bridges with strong Szego limit theorem

TL;DR

This paper introduces a novel method for calculating four-point correlation functions in planar N=4 SYM theory using Fredholm determinants and Szego limit theorem, applicable at both weak and strong coupling regimes.

Contribution

It presents a new technique expressing octagons as Fredholm determinants of Bessel operators, enabling efficient calculations across coupling strengths and generalizing asymptotic determinant results.

Findings

Efficient computation of octagons at weak and strong coupling.

Derivation of the leading asymptotic behavior using Szego limit theorem.

Development of a systematic approach for subleading corrections.

Abstract

We develop a new technique for computing a class of four-point correlation functions of heavy half-BPS operators in planar N=4 SYM theory which admit factorization into a product of two octagon form factors with an arbitrary bridge length. We show that the octagon can be expressed as the Fredholm determinant of the integrable Bessel operator and demonstrate that this representation is very efficient in finding the octagons both at weak and strong coupling. At weak coupling, in the limit when the four half-BPS operators become null separated in a sequential manner, the octagon obeys the Toda lattice equations and can be found in a closed form. At strong coupling, we exploit the strong Szego limit theorem to derive the leading asymptotic behavior of the octagon and, then, apply the method of differential equations to determine the remaining subleading terms of the strong coupling expansion to any order in the inverse coupling. To achieve this goal, we generalize results available in the literature for the asymptotic behavior of the determinant of the Bessel operator. As a byproduct of our analysis, we formulate a Szego-Akhiezer-Kac formula for the determinant of the Bessel operator with a Fisher-Hartwig singularity and develop a systematic approach to account for subleading power suppressed contributions.