Heavy tails in Calabi-Yau moduli spaces

Abstract

We study the statistics of the metric on Kähler moduli space in compactifications of string theory on Calabi-Yau hypersurfaces in toric varieties. We find striking hierarchies in the eigenvalues of the metric and of the Riemann curvature contribution to the Hessian matrix: both spectra display heavy tails. The curvature contribution to the Hessian is non-positive, suggesting a reduced probability of metastability compared to cases in which the derivatives of the Kähler potential are uncorrelated. To facilitate our analysis, we have developed a novel triangulation algorithm that allows efficient study of hypersurfaces with h1,1 as large as 25, which is difficult using algorithms internal to pack-ages such as Sage. Our results serve as input for statistical studies of the vacuum structure in flux compactifications, and of the distribution of axion decay constants in string theory.