# The Octagon as a Determinant

## TL;DR

This paper derives a non-perturbative determinant formula for the octagon form factor related to four-point functions of BPS operators, connecting it to weak coupling expansions and free field representations.

## Contribution

It introduces a non-perturbative determinant expression for the octagon and links it to perturbative ladder diagrams and free boson/fermion operator representations.

## Key findings

- Non-perturbative formula for the octagon as a determinant.
- Weak coupling limit expressed via polylogarithm-based determinants.
- Operator representation using free bosons or fermions.

## 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.

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Source: https://tomesphere.com/paper/1905.11467