Andrew Harder - Motivic geometry of 2-loop Feynman integrals - 14.06.2023
Channel: Hörsaal Mathematikon
Date: 14/06/2023

Hodge theory, Mirror Symmetry, and Physics of Calabi-Yau Moduli
https://web.mathi.uni-heidelberg.de/cymod23
A MATCH-PIMS-STRUCTURES workshop in Mathematical Physics, held from June 12 to 16, 2023, at the Mathematikon in Heidelberg
For more talks visit https://www.youtube.com/playlist?list=PL57PsbjMibmQ37JSKzyxqvmIhyAfF1iFs

Abstract: 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.