Counting String Theory Standard Models

Abstract

We derive an approximate analytic relation between the number of consistent heterotic Calabi-Yau compactifications of string theory with the exact charged matter content of the standard model of particle physics and the topological data of the internal manifold: the former scaling exponentially with the number of Kahler parameters. This is done by an estimate of the number of solutions to a set of Diophantine equations representing constraints satisfied by any consistent heterotic string vacuum with three chiral massless families, and has been computationally checked to hold for complete intersection Calabi-Yau threefolds (CICYs) with up to seven Kahler parameters. When extrapolated to the entire CICY list, the relation gives about 10^23 string theory standard models; for the class of Calabi-Yau hypersurfaces in toric varieties, it gives about 10^723 standard models.