Philip Candelas (Oxford) - The Arithmetic of Calabi-Yau Manifolds, Black Holes and Mirror Symmetry

Title: The Arithmetic of Calabi-Yau Manifolds, Black Holes and Mirror Symmetry

Abstract: The main aim of this talk is to explain that there are questions of common interest, in the context of the arithmetic of Calabi Yau 3-folds, to physicists, number theorists and geometers. The main quantities of interest, in the arithmetic context, are the numbers of points of the manifold considered as a variety over a finite field. We are interested in the computation of these numbers and their dependence on the moduli of the variety. The surprise, for a physicist, is that the numbers of points over a finite field are also given by expressions that involve the periods of a manifold. The number of points are encoded in the local zeta function, about which much is known in virtue of the Weil conjectures. I will discuss a number of topics related to the zeta function, mirror symmetry and the appearance of modularity for one parameter families of Calabi-Yau manifolds. I will report on an example for which the quartic numerator of the zeta function factorises into two quadrics at special values of the parameter, which satisfy an algebraic equation with coefficients in Q (so independent of any particular prime), and for which the underlying manifold is smooth. We note that these factorisations are due to a splitting of the Hodge structure and that these special values of the parameters are rank two black hole attractor points in the sense of type IIB supergravity. Modular groups and modular forms arise in relation to these attractor points. To our knowledge, the rank two attractor points that were found by the application of these number theoretic techniques, provide the first explicit examples of such points for Calabi-Yau manifolds of full SU(3) holonomy. Many interesting identities follow from this identification of a rank two attractor point, some of these lead to intriguing identities for the mirror manifold.

This talk reports on joint work with Mohamed Elmi, Xenia de la Ossa and Duco van Straten.