Quadratic relations between Bessel moments

TL;DR

This paper proves quadratic relations between Bessel moments related to Feynman amplitudes by interpreting them as period pairings in algebraic geometry, confirming conjectured relations and linking to special values of L-functions.

Contribution

It provides a geometric interpretation of Bessel moments as period pairings, proving the conjectured quadratic relations and connecting them to Deligne's conjecture on L-values.

Findings

Proved quadratic relations between Bessel moments.

Connected Bessel moments to period pairings in algebraic geometry.

Explicitly related evaluations to critical L-values.

Abstract

Motivated by the computation of certain Feynman amplitudes, Broadhurst and Roberts recently conjectured and checked numerically to high precision a set of remarkable quadratic relations between the Bessel moments \[ \int_0^\infty I_0(t)^i K_0(t)^{k-i}t^{2j-1}\,\mathrm{d}t \qquad (i, j=1, \ldots, \lfloor (k-1)/2\rfloor), \] where $k \geq 1$ is a fixed integer and $I_0$ and $K_0$ denote the modified Bessel functions. In this paper, we interpret these integrals and variants thereof as coefficients of the period pairing between middle de Rham cohomology and twisted homology of symmetric powers of the Kloosterman connection. Building on the general framework developed in arXiv:2005.11525, this enables us to prove quadratic relations of the form suggested by Broadhurst and Roberts, which conjecturally comprise all algebraic relations between these numbers. We also make Deligne's conjecture explicit, thus explaining many evaluations of critical values of $L$-functions of symmetric power moments of Kloosterman sums in terms of determinants of Bessel moments.