The Ricci Curvature of Half-flat Manifolds

TL;DR

This work derives explicit expressions for the Ricci curvature $R_{mn}$ and scalar $R$ of six-dimensional half-flat $SU(3)$-structure manifolds by embedding them in seven-dimensional manifolds with $G_2$ holonomy, thereby linking intrinsic torsion to extrinsic curvature. The main results express $R_{mn}$ and $R$ in terms of torsion classes $W_i$ and are validated on Iwasawa and general nilmanifolds; the authors further specialize to half-flat manifolds arising from mirror symmetry with NS fluxes and extract a constraint on the Kähler moduli via a consistency condition, together with Hitchin-flow for the internal volume. In the mirror-symmetry setting, they provide the moduli-space language expressions for torsion and curvature, derive the corresponding superpotential structure, and demonstrate how flux data constrains the geometry through a simple, elegant relation between the flux-induced charges and the Calabi–Yau moduli. Overall, the results connect half-flat geometry to low-energy effective actions in Type II compactifications and offer new avenues for moduli stabilization within generalized Calabi–Yau frameworks.

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 Kähler moduli space of type II string theories on these half-flat manifolds.