Seminar "Automorphic Forms and Applications". Speaker : Pierre Vanhove (Institut de Physique Theorique of CEA, France)

Abstract:

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.