At Moller Institute: The arithmetic of families of Calabi-Yau manifolds: black holes and modularity

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• Philip Candelas (University of Oxford), Xenia de la Ossa (University of Oxford)
• Tuesday 16 August 2022, 18:00-19:00
• No Room Required.

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NCN2 - New connections in number theory and physics

The main goal of this talk is to explore questions of common interest for physicists, number theorists and geometers,  in the context of the arithmetic of Calabi Yau 3-folds. 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. In these talks 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 discuss  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 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 parameter 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 attractor points for Calabi-Yau manifolds.  Time permitting, we will describe this scenario also for the mirror manifold in type IIA supergravity.

This talk is part of the Isaac Newton Institute Seminar Series series.

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