The Arithmetic of Calabi–Yau Manifolds

In the research area of string compactifications and, more recently, in the context of quantum field theory scattering amplitudes, geometric properties of Calabi–Yau varieties and their moduli spaces have played an important role. In the past these Calabi–Yau varieties have often been studied over the field of complex numbers. Considering such varieties over the field of rational numbers, or over finite fields, is not only a modern research topic in arithmetic geometry and number theory, but also offers novel perspectives and new insights in physics. For instance: the arithmetic of Calabi–Yau manifolds is
linked to supersymmetric black holes and supersymmetric flux vacua of string theory, the asymptotic growth of Gopakumar–Vafa invariants is determined by the L-values of modular Calabi–Yau varieties, and similar arithmetic methods have been applied in the context of quantum field theory Feynman integrals.