Complex Multiplication of Exactly Solvable Calabi-Yau Varieties

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.