Non-Kaehler String Backgrounds and their Five Torsion Classes

TL;DR

This paper develops a systematic framework for six-dimensional non-Kähler manifolds with SU(3) structure in flux backgrounds, establishing a dictionary between four-dimensional ${\cal N}=1$ heterotic SUSY conditions and the intrinsic torsion classes ${\cal W}_1$ through ${\cal W}_5$. It analyzes how nonzero H-flux deforms Calabi–Yau geometries into constrained, non-Kähler spaces, and demonstrates a concrete heterotic background on the Iwasawa manifold that satisfies all SUSY and Bianchi constraints using Abelian gauge flux. The work also connects these torsion classifications to M-theory via G2 structures and outlines future directions for extending the framework to conformally balanced manifolds, action-based derivations, and G2 uplifts. Overall, it provides a detailed map between geometry and flux backreaction in string compactifications and identifies explicit solvable examples. The results have implications for moduli stabilization, flux-induced potentials, and the construction of realistic heterotic backgrounds.

Abstract

We discuss the mathematical properties of six--dimensional non--Kähler manifolds which occur in the context of ${\cal N}=1$ supersymmetric heterotic and type IIA string compactifications with non--vanishing background H--field. The intrinsic torsion of the associated SU(3) structures falls into five different classes. For heterotic compactifications we present an explicit dictionary between the supersymmetry conditions and these five torsion classes. We show that the non--Ricci flat Iwasawa manifold solves the supersymmetry conditions with non--zero H--field, so that it is a consistent heterotic supersymmetric groundstate.