Modularity of Calabi-Yau Manifolds and String Theory Compactifications

Calabi-Yau compactifications of superstring theory yield a surprisingly varied set of theories, ranging from cosmological flux vacua to supersymmetric black holes. It turns out that in many cases the physical properties of these theories are related to intricate arithmetic properties of the compactification manifold. In this talk, I discuss how techniques from number theory and arithmetic geometry can be used to find some of these solutions and compute many of their relevant physical quantities, concentrating mostly on the relation between modularity of Calabi-Yau manifolds and supersymmetric Minkowski flux vacua. I introduce briefly the zeta function of a manifold, discuss how it is computed and how it can be used to find these vacua. In special cases, the zeta function is related to (elliptic) modular forms, and conjectures in number theory relate the modular forms to physical quantities, such as the vacuum expectation value of the background fields. I present a simple method that can be used to construct many families of supersymmetric flux vacua for which we have explicitly tested these conjectures. The talk is based on several works together with Candelas, de la Ossa, McGovern, Jockers, and Kotlewski.