The Ricci Curvature of Half-flat Manifolds

TL;DR

This paper derives formulas for Ricci curvature of half-flat manifolds using their intrinsic torsion classes, linking geometry with string theory compactifications and mirror symmetry.

Contribution

It provides explicit Ricci curvature expressions for half-flat manifolds and explores their implications in string theory moduli spaces and flux compactifications.

Findings

Derived Ricci curvature formulas in terms of torsion classes.

Tested formulas on Iwasawa and nilpotent manifolds.

Established constraints on Kähler moduli space in string theory.

Abstract

We derive expressions for the Ricci curvature tensor and scalar in terms of intrinsic torsion classes of half-flat manifolds by exploiting the relationship between half-flat manifolds and non-compact $G_2$ holonomy manifolds. Our expressions are tested for Iwasawa and more general nilpotent manifolds. We also derive expressions, in the language of Calabi-Yau moduli spaces, for the torsion classes and the Ricci curvature of the \emph{particular} half-flat manifolds that arise naturally via mirror symmetry in flux compactifications. Using these expressions we then derive a constraint on the K\"ahler moduli space of type II string theories on these half-flat manifolds.