Arithmetic and Attractors

TL;DR

This paper explores deep connections between number theory and supersymmetric black holes through the attractor mechanism, revealing that attractor varieties relate to elliptic curves with complex multiplication and class fields, with implications for Calabi-Yau compactifications.

Contribution

It demonstrates that attractor varieties in N=2 supergravity are constructed from elliptic curves with complex multiplication and links these to class fields and modular forms, extending the attractor mechanism to arithmetic geometry.

Findings

Attractor varieties are built from products of elliptic curves with complex multiplication.

U-duality inequivalent backgrounds with same horizon area relate to class numbers of quadratic imaginary fields.

Connections to Kronecker's Jugendtraum and modular heights are discussed.

Abstract

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).