Attractors and Arithmetic

TL;DR

This paper explores the connection between attractor varieties in string theory and elliptic curves with complex multiplication, revealing deep links between arithmetic geometry and black hole solutions in Calabi-Yau compactifications.

Contribution

It demonstrates that attractor varieties in certain string compactifications are constructed from elliptic curves with complex multiplication, highlighting new links between arithmetic geometry and string theory.

Findings

Attractor varieties are products of elliptic curves with complex multiplication.

Heterotic dual theories relate to rational conformal field theories.

Connections between arithmetic and string theory are suggested.

Abstract

We consider attractor varieties arising in the construction of dyonic black holes in Calabi-Yau compactifications of IIB string theory. We show that the attractor varieties are constructed from products of elliptic curves with complex multiplication for $\mathcal{N}=4,8$ compactifications. The heterotic dual theories are related to rational conformal field theories. The emergence of curves with complex multiplication suggests many interesting connections between arithmetic and string theory. This paper is a brief overview of a longer companion paper entitled ``Arithmetic and Attractors,'' hep-th/9807087.