Heavy Tails in Calabi-Yau Moduli Spaces

TL;DR

This work analyzes the statistics of the Kahler moduli-space metric in Calabi–Yau compactifications and its curvature contributions to the Hessian, revealing heavy-tailed, highly hierarchical spectra that depart from naive i.i.d. randomization models. By constructing an ensemble of CY Kahler metrics via a novel triangulation of reflexive polytopes and computing the curvature Hessian $igl( ext{H}_Rigr)$, the authors show that $igl( ext{H}_Rigr)$ is non-positive and exhibits pronounced negative tails, especially near walls of the Kahler cone, which has implications for metastability and axion-decay constants. The results highlight the importance of geometric correlations in Kahler potentials, challenge the conventional i.i.d. random supergravity framework, and point to refined models (e.g., Bergman metrics) that better capture CY-like spectra. Overall, the findings impact statistical vacuum studies and the distribution of axion decay constants in string theory, with potential guidance for model-building in flux compactifications and beyond.

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 $h^{1,1}$ 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.

Figures (28)

• Figure 1: Eigenvalue distributions for two classical matrix ensembles. The data displayed result from collections of 1000 matrices with $200\times 200$ i.i.d. entries with $\sigma=1/\sqrt{200}$.
• Figure 2: Eigenvalue distributions for random matrix models of the i.i.d. Hessian $\mathcal{H}^{\mathrm{i.i.d.}}\approx \mathcal{H}^{\mathrm{WWW}}$\ref{['eq:RMT_Hess']} (taken with $3\left\lvert W\right\rvert^{2}/F^{2}=.01$) and of the Riemann contribution $\mathcal{H}_{R}^{\mathrm{i.i.d.}}\approx \mathcal{H}^{\mathrm{WW}}_{R}$\ref{['eq:WW_curvature_Hessian_NH']}. In both cases, the spectra are in units of $\left\lvert F\right\rvert^{2}$. Figure \ref{['fig:wasteland']} is adapted from Figure 3 of Marsh:2011aa.
• Figure 3: A non-simplicial cone in three dimensions. The cone is generated by the five rays originating from the apex.
• Figure 4: The nine smallest eigenvalues, coded by color, of each of $10^{5}$ metrics on the Kähler moduli space of a threefold with $h^{1,1}=15$.
• Figure 5: The largest eigenvalues and the entire spectrum of the metric on the Kähler moduli space of a threefold with $h^{1,1}=15$. In (a), the tail of the distribution was truncated to demonstrate the bulk of the distribution. The entire spectrum has support from $2.6\times 10^{-5}$ to $1.2\times 10^{9}$.