Description
In this talk, I first give a brief overview of the connection between Feynman
integrals and Calabi-Yau geometries. Then I present a novel construction of a family of
Abelian curves whose periods are associated to Feynman integrals. To find such curves,
we first use the connection between Feynman integrals and Calabi-Yau manifolds to
express the integrals in terms of Calabi-Yau periods. Natural candidates for Abelian curves
encoding these periods are then given by intermediate Jacobians, which are well-studied
tori that can be associated to any Calabi-Yau manifolds. The classical (Griffiths and Weil)
Jacobians are either Abelian or vary holomorphically, but not both, and thus do not provide
a simple relation between Feynman integrals and periods. This motivates us to construct a
novel type of intermediate Jacobian with both of the desirable properties. The price one
has to pay is that the Jacobian is not defined everywhere in the moduli space of the
Calabi-Yau manifold. However, this restriction is very natural from the amplitudes
perspective: the moduli of the Calabi-Yau manifold are already restricted by conditions
such as that the masses appearing in the corresponding Feynman integrals are real.
This talk is based on joint work with Jockers, Kotlewski, McLeod, Pögel, Sarve, Wang, and
Weinzierl.