Feynman Integrals, modular forms and beyond

In this talk I will explain the link between Feynman integrals and period of mixed Hodge structures. We will discuss the case of the sunset graph which are associated to Calabi-Yau geometry and show how the Feynman integral is expressed as modular forms at low loop order. We will present a very intriguing link with mirror symmetry and the conjecture that the sunset graph Feynman integral compute the prepotential of local Gromov-Witten invariant of some non-compact Calabi-Yau associate to the graph polynomial.