Towers of Feynman Integrals

In this talk we describe the geometry of the family of multiloop sunset graph hypersurfaces. We will show that they are described by a family of Calabi-Yau n-fold $X_n$ where the $X_n$ is elliptically fibered over $X_{n-1}$. We will describe the elliptic fibration.
We show that the graph hypersurface has a determintal representation. We will detail the case of the 3-loop graph hypersurface which defines a K3 surface given by the Hessian quartic K3 surfaces. We will detail the lattice polarisation and show that one needs to refine the general theory.
We will then discuss the four-loop sunset which is given by the small projective resolution of the 30 nodal Calabi-Yau threefolds after Hulek-Verrill. This is based on work in progress with Charles Doran and Andrey Novoseltsev.