21st Geometry and Physics Seminar

Title: Black Holes, arithmetic and modularity for families of
Calabi-Yau manifolds

The seminars are jointly organized by the Department of Mathematics and the Department of Physics and Astronomy, coffee and tea will be served after the first talk at 16:15.

Abstract: The main goal of these two talks is to explore some questions of common interest for physicists, number theorists and geometers, in the context of the arithmetic of Calabi-Yau 3-folds. There are many such relations, however we will focus on the rich structure of black hole solutions of type II superstrings on a Calabi-Yau manifolds. We will give a self contained introduction aimed at a mixed audience of physicists and mathematicians. 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. A mathematician is 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. These periods determine many aspects of the physical theory, as for example the kinetic terms of the effective Lagrangian as well as the Yukawa couplings, but also properties of black hole solutions. For a mathematician, the number of points determine the zeta function, about which much is known in virtue of the Weil conjectures. We discuss a number of interesting topics related to the zeta function, the corresponding L-function, and the appearance of modularity for one parameter families of Calabi-Yau manifolds. We will focus on an example for which the quartic numerator of the zeta function of a one parameter family factorises into two quadrics at special values of the parameter, which satisfies an algebraic equation with coefficients in Q (so independent of any particular prime), and for which the underlying manifold is smooth. The significance of these factorisations in physics is that they are due to the existence of black hole attractor points in the sense of type II supergravity and are related to a splitting of the Hodge structure and that at these special values of the parameter. For a mathematician these factorisations of the Hodge structure are related to the famous Hodge Conjecture. 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 attractor points for Calabi-Yau manifolds. Time permitting, we will describe this scenario also for the mirror manifold in type IIA supergravity.