Feynman Motives

California Institute Of Technology, Pasadena CA

Investigators

Abstract

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.