Massively parallel computations in algebraic geometry

Design and implementation of multi-threaded algorithms in polynomial algebra

Differential equations for Feynman integrals

Abstract:

Feynman integrals enter the evaluation of many physical observables in particle physics, gravitational physics, statistical physics and solid-state physics. They are multidimensional integrals which cannot be evaluated with elementary methods. It is an important question to understand what kind of special functions they are of their physical parameters.

In this tutorial we will present algebraic geometrical approaches for determining the Feynman integral D-modules. We will first introduce the parametric representations of a generic Feynman integral, and explain that Feynman integrals are holonomic functions, and that their ideal of annihilators generates a holonomic D-module. Then we will present a derivation of the holonomic D-module using the creative telescoping algorithm. We will then analyse the relation between this D-module and the generalized Gel'fand-Kapranov-Zelevinsky D-module obtained from a toric geometrical approach to the Feynman integrals. This tutorial will be illustrated by several examples.