Feynman integrals, L-series and Kloosterman moments

TL;DR

This paper explores deep connections between Feynman integrals, L-series, and Kloosterman sums, proposing new conjectures and algorithms that reveal intricate structures linking quantum field theory, algebraic geometry, and number theory.

Contribution

It introduces new conjectures and algorithms relating Feynman integrals to L-series derived from Kloosterman moments, including evaluations of modular forms and determinants up to 30 loops.

Findings

Conjectural evaluations of non-critical L-series of modular forms of weights 3, 4, and 6.

Formulas for determinants of Feynman integrals tested up to 30 loops.

Identification of features in L-series structure not determined by the functional equation alone.

Abstract

This work lies at an intersection of three subjects: quantum field theory, algebraic geometry and number theory, in a situation where dialogue between practitioners has revealed rich structure. It contains a theorem and 7 conjectures, tested deeply by 3 optimized algorithms, on relations between Feynman integrals and L-series defined by products, over the primes, of data determined by moments of Kloosterman sums in finite fields. There is an extended introduction, for readers who may not be familiar with all three of these subjects. Notable new results include conjectural evaluations of non-critical L-series of modular forms of weights 3, 4 and 6, by determinants of Feynman integrals, an evaluation for the weight 5 problem, at a critical integer, and formulas for determinants of arbitrary size, tested up to 30 loops. It is shown that the functional equation for Kloosterman moments determines much but not all of the structure of the L-series. In particular, for problems with odd numbers of Bessel functions, it misses a crucial feature captured in this work by novel and intensively tested conjectures. For the 9-Bessel problem, these lead to an astounding compression of data at the primes.