Abstract
We study the statistics of the metric on K\"ahler 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 K\"ahler potential are uncorrelated. To facilitate our analysis, we have developed a novel triangulation algorithm that allows efficient study of hypersurfaces with $h^{1,1}$ as large as 25, which is difficult using algorithms internal to packages 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.
Highlights
The vacuum structure of compactifications of string theory to four dimensions is extremely complex, and direct enumeration of all vacua appears impossible
We study the statistics of the metric on Kahler moduli space in compactifications of string theory on Calabi-Yau hypersurfaces in toric varieties
We obtain a related result for the Kahler metrics themselves: the eigenvalue spectra of the metrics on Kahler moduli spaces exhibit large hierarchies and heavy tails, at least in the case of O3/O7 orientifolds that we study directly
Summary
The vacuum structure of compactifications of string theory to four dimensions is extremely complex, and direct enumeration of all vacua appears impossible. It is plausible that the number of vacua of string theory that are consistent with all present observations vastly exceeds the number of measurements of fundamental parameters that could be performed in the future. In this situation it is impractical, and for many purposes misguided, to pursue individual vacua of string theory as candidate models of the observed universe. Quantum corrections to the effective action are poorly understood, and many of the most incisive results concern the distribution of supersymmetric Minkowski vacua of a subsector of the theory, in the classical approximation. The elegant result of [5] in type IIB flux vacua describes an index approximating the distribution of configurations for which the classical
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