Javier FresánClaude SabbahJENG-DAW YU2023-06-052023-06-052020-06-0419370652https://scholars.lib.ntu.edu.tw/handle/123456789/63177440 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 tableMotivated 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.Bessel moments | Kloosterman connection | period pairing | quadratic relations; Mathematics - Algebraic Geometry; Mathematics - Algebraic Geometry; Mathematics - Number Theoryjournal article10.2140/ant.2023.17.5412-s2.0-85159720914http://arxiv.org/abs/2006.02702v2