Quadratic relations between Bessel moments

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 $k\ge 1$ 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.

arXiv comment40 pages, 1 figure, 1 table. V2: Comparison of periods now takes place in the setting of exponential mixed Hodge structures. Add an appendix by the second author for necessary tools. Change format. 61 pages, 1 figure, 1 table

References in corpus (8)

• A Feynman integral via higher normal functions
• Periods for flat algebraic connections
• Hodge theory of Kloosterman connections
• Wrońskian factorizations and Broadhurst-Mellit determinant formulae
• Wick rotations, Eichler integrals, and multi-loop Feynman diagrams
• Hilbert Transforms and Sum Rules of Bessel Moments
• Wrońskian algebra and Broadhurst-Roberts quadratic relations
• $\mathbb{Q}$-linear dependence of certain Bessel moments

Cited by in corpus (9)

• Decomposition of Feynman Integrals by Multivariate Intersection Numbers
• Hodge theory of Kloosterman connections
• Quadratic relations between periods of connections
• Twisted Riemann bilinear relations and Feynman integrals
• $\mathbb{Q}$-linear dependence of certain Bessel moments
• Hodge properties of Airy moments
• Co-homology of Differential Forms and Feynman diagrams
• On the Periods of Twisted Moments of the Kloosterman Connection
• Singularities of functions: a global point of view