Vector Spaces and Differential Equations for Feynman Integrals and Beyond

Feynman integrals, are the building blocks of scattering amplitudes in Quantum Field Theories, and are essential for bridging theoretical predictions with experimental results. They appear across scales in modern physics —— from particle collisions at the Large Hadron Collider, to black hole mergers at gravitational wave detectors at LIGO and VIRGO. Understanding the properties of Feynman integrals is thus crucial for the evaluation of scattering amplitudes, with far reaching experimentally relevant consequences. This thesis examines the interplay between Quantum Field Theory and computational algebraic and differential geometry, focusing on interpreting Feynman integrals as elements of the twisted de Rham cohomology vector space. This formalism cleanly reformulates concepts originally developed via integration by parts techniques. Feynman integrals are interpreted as vectors, master integrals serve as basis vectors, and the number of masters is given by the vector space’s dimension. Through the usage of an inner product, known as the (twisted cohomology) intersection number, Feynman integral decompositions are achieved via projections onto the master integral basis. This provides a conceptually clear method for reducing amplitudes, as well as constructing differential equations, in a complimentary way to integration by parts techniques. In this thesis, we explore many of the latest advancements in the field of intersection theory, develop new methods for the evaluation of intersection numbers, and present their applications to one- and two-loop Feynman and Fourier integrals. We furthermore explore the relations between ordinary twisted cohomology and relative twisted cohomology intersection numbers, by analysing special properties of analytic regulators, which are present in the former type and absent in the latter. Specifically we demonstrate how the relative intersection number, built through the usage of delta forms, can be extracted from the analytically regulated intersection number through a careful limiting procedure. We additionally discuss how polynomial division techniques over finite fields can be used to simplify the evaluation of intersection numbers, avoiding algebraic field extensions. Special basis choices, as well as previously unknown patterns in intersection numbers are also discussed in detail. The analytic evaluation of master Feynman integrals is also considered in this work. Through the usage of the differential equations method, and a canonical basis obtained via the Dyson/Magnus exponential matrix, we provide, for the first time, analytic expressions for a set of two-loop three-point functions contributing to massive-particle form factors. The renormalisation is performed via modern diagrammatic techniques. We finally showcase how the techniques developed for and applied to Feynman integrals can be as well applied to a much broader class of integrals, admitting a twisted period representation. Using the method of intersection theory and differential equations, we find for the first time analytic solutions to Fourier integrals relevant to various areas of Field Theory in the quantum as well as classical contexts. The result of this thesis are relevant to both physical and mathematical applications.