Calabi-Yau Manifolds and Feynman Integral Computations

Abstract

In this thesis we use geometrical and string theoretic inspired methods to compute Feynman integrals. We analyze the important family of l-loop banana Feynman graphs. For this we relate the abstract l-loop Feynman integral in D=2 dimensions to geometric period integrals of a l-1-dimensional Calabi-Yau manifold such that the maximal cut contours correspond to the integral homology. Quadratic relations between banana Feynman integrals are derived from Griffiths transversality. The monodromy behavior of the banana integral at large momentum together with special properties of Calabi-Yau manifolds is used to completely determine the banana Feynman integral for large momentum. First, we give an introduction to basics of Feynman integral computations, fundamentals of the theory of linear differential equations and the mathematics of Calabi-Yau spaces in the context of Feynman graphs. Next, we use these concepts and techniques to compute the banana Feynman integrals to high loop orders in the equal-- as well as in the generic-mass case.