Complex Multiplication of Exactly Solvable Calabi-Yau Varieties

TL;DR

This paper introduces a framework linking exactly solvable Calabi-Yau varieties to abelian varieties with complex multiplication, revealing deep number-theoretic and geometric structures underlying string theory models.

Contribution

It provides a novel conceptual approach connecting Calabi-Yau solvability to complex multiplication on abelian varieties derived from their cohomology.

Findings

Calabi-Yau solvability characterized by complex multiplication

Conformal field theory quantities linked to number theoretic structures

Discrete geometric structures explained via torsion points on abelian varieties

Abstract

We propose a conceptual framework that leads to an abstract characterization for the exact solvability of Calabi-Yau varieties in terms of abelian varieties with complex multiplication. The abelian manifolds are derived from the cohomology of the Calabi-Yau manifold, and the conformal field theoretic quantities of the underlying string emerge from the number theoretic structure induced on the varieties by the complex multiplication symmetry. The geometric structure that provides a conceptual interpretation of the relation between geometry and the conformal field theory is discrete, and turns out to be given by the torsion points on the abelian varieties.