Arithmetic and Attractors

We study relations between some topics in number theory and supersymmetric black holes. These relations are based on the attractor mechanism'' of N=2 supergravity. In IIB string compactification this mechanism singles out certain attractor varieties.‘’ We show that these attractor varieties are constructed from products of elliptic curves with complex multiplication for N=4 and N=8 compactifications. The heterotic dual theories are related to rational conformal field theories. In the case of N=4 theories U-duality inequivalent backgrounds with the same horizon area are counted by the class number of a quadratic imaginary field. The attractor varieties are defined over fields closely related to class fields of the quadratic imaginary field. We discuss some extensions to more general Calabi-Yau compactifications and explore further connections to arithmetic including connections to Kronecker’s Jugendtraum and the theory of modular heights. The paper also includes a short review of the attractor mechanism. A much shorter version of the paper summarizing the main points is the companion note entitled ``Attractors and Arithmetic’’ (hep-th/9807056).