Arithmetic & Algebraic Geometry Seminar: "Motivic geometry of 2-loop Feynman integrals"

Abstract: Feynman integrals are quantities which appear frequently when one computes amplitudes in quantum field theory. As a result of work of Bloch-Esnault-Kreimer, and subsequent work of Brown, one can reinterpret Feynman integrals as periods of mixed Hodge structures attached to pairs of hypersurfaces in certain toric varieties. The hypersurfaces taking part in this construction are built from a Feynman graph. This is essentially the data of a graph decorated with mass parameters attached to each edge and momentum parameters attached to each vertex.  Despite the combinatorial origin of their geometry, the motives of Bloch-Esnault-Kreimer and Brown are only understood in a few basic cases. In this talk, I will discuss results which show that, for a simple, but infinite, class of graphs, the mixed Hodge structures controlling the corresponding Feynman integrals are built from hyperelliptic curves. This generalizes results of Bloch and Kerr and helps clarify many computations appearing in the physics literature. I'll demonstrate this in some examples and discuss the relationship to computational work of Lairez-Vanhove.

This is joint work with C. Doran and P. Vanhove.

Host: Matt Kerr