Bounding the Complexity of Feynman Integrals with Hodge Theory

Twenty years ago, Bloch, Esnault, and Kreimer introduced an algebro-geometric formulation of Feynman integration, building on Griffiths’ theory of variation of mixed Hodge structure. Explicit computation for specific Feynman graphs with all parameters has proved an elusive goal. With Andrew Harder and Pierre Vanhove, we use quadric bundles to establish a complexity bound on the motives underlying an infinite collection of two-loop Feynman integrals. For another family of graphs with unbounded loop order, we describe the geometry and Hodge theory of the Feynman motives of Calabi-Yau type.