Quantum field theory is the most sophisticated technique for predictive computations in high-energy and particle physics. The use of Feynman diagrams as computational devices makes it possible to obtain high precision computations of physical processes involving elementary particles and quantum fields. Despite its long history of successful applications to the world of particle physics, the mathematics of quantum field theory is still mysterious and full of beautiful challenges and open problems. Numerical evidence suggests that the procedure of extracting finite values from divergent Feynman integrals gives rise to a class of numbers, multiple zeta values, that are of great significance to number theory and algebraic geometry. This suggests a mysterious relation between quantum field theory and an important research topic of current interest in pure mathematics: Grothendieck's theory of motives of algebraic varieties. The purpose of this research proposal is to understand the nature of this relation and investigate what results can be derived from it, both in terms of gaining some better understanding of the very difficult multi-loop computations of Feynman integrals using tools from algebraic geometry, and conversely of understanding how we can extend our current knowledge of motives using quantum field theory.
One of the main questions under investigation is when, possibly after a subtraction of divergences, the computation of a Feynman integral for a scalar quantum field theory results in a period of a mixed Tate motive. Using the Feynman parametric form, this question reflects the motivic nature of a relative cohomology of an affine hypersurface constructed out of the data of the Feynman integral. The subtraction of divergences is encoded in a Hopf algebra structure, which is itself related to Hopf algebras and dual groups that appear naturally in the theory of motives. One of the main steps that are needed to further understand the relation between quantum field theory and motives is combining the more concrete approach via the algebraic geometry of hypersurfaces of Feynman graphs with the more abstract approach via Tannakian categories and Hopf algebras.
In high energy physics, a powerful method for computing outcomes of particle collisions and interactions is provided by perturbative quantum field theory, where the computation is broken into the contribution of a series of Feynman diagrams, illustrating all the various possible creations and annihilations of virtual particles that take place during such high energy events. The computations associated to Feynman diagrams can become extraordinarily difficult, as soon as the complexity of the diagram grows. This calls for the development of new mathematical methods aimed at understanding the structure of these Feynman diagram calculations and possibly simplify them, by identifying the main organizing principles. In recent years, both mathematicians and theoretical physicists studying this question have noticed the mysterious occurrence, in Feynman integral calculations, of a class of numbers (called multiple zeta values) that are well known to mathematicians working in the abstract fields of number theory and arithmetic geometry. These numbers belong to a larger class, called periods, which are closely related to one of the fundamental developments of 20th century mathematics: Grothendieck's theory of motives of algebraic varieties. This very abstract theory has never before found applications outside of pure mathematics, a fact that makes its unexpected occurrence in high energy physics extremely striking. The PI was one of the researchers who first inititated this whole line of investigation: she was one of the first scientists who noticed the existence of a deep connection between these two theories. The work conducted under this award is a systematic investigation of the relation between motives and high energy physics, aimed at better understanding this surprising connection and use it both to extend our understanding of motives, and to better control the intricacies of the Feynman integral computations.
