Feynman integrals, L-series and Kloosterman moments

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.

Rationalizing loop integration

“…It is hoped that further study of these examples will (alongside similar examples [32]) lead to a more systematic understanding of how these barriers can be resolved more generally. In cases where Feynman-parameter-dependent square roots can no longer be avoided, elliptic integrals [24,[79][80][81][82] and integrals of even higher complexity occur [25,83,84]. These will require a new perspective on direct integration, rooted in a deeper understanding of the functions required.…”

Feynman Integrals, Toric Geometry and Mirror Symmetry

“…Which gives the dual third period R 1 , such that π (2) 1 (p 2 , ξ 2 ) = d d p 2 R 1 (p 2 , ξ 2 ). This dual period R 1 is therefore identified with the derivative of local prepotential F 0 11 It has been already noticed in [74] the special role played by the Mahler measure and mirror symmetry. 12 We would like to thank Albrecht Klemm for discussions and communication that helped clarifying the link between the work in [20] and the analysis in [24].…”

Some Algebraic and Arithmetic Properties of Feynman Diagrams

“…In perturbative quantum field theory (pQFT), we use Feynman diagrams to quantify the interactions among elementary particles [31,1,11,37]. In this survey, we will focus on 2-dimensional pQFT, where the propagator of a free particle with proper mass m 0 takes the following form: 1 (2π) 2 lim ε→0 + R 2 e ip p p·x x x−ε|p p p| 2 d 2 p p p |p p p| 2 + m 2 0 = K 0 (m 0 |x x x|) 2π (1) for x x x ∈ R 2 {0 0 0}.…”