Local descriptions of the heterotic SU(3) moduli space

TL;DR

This paper analyzes the local structure of the moduli space of solutions to the heterotic SU(3) system on a compact 6-manifold, revealing it has expected zero dimension and establishing links between different cohomology theories.

Contribution

It introduces a deformation complex for the heterotic SU(3) system, proving the moduli space has expected zero dimension and connecting cohomology groups via a Dolbeault-type theorem.

Findings

Moduli space has expected dimension zero.

Isomorphism between deformation and obstruction cohomology groups.

Dolbeault-type theorem linking cohomology to Čech cohomology.

Abstract

The heterotic $SU(3)$ system, also known as the Hull--Strominger system, arises from compactifications of heterotic string theory to six dimensions. This paper investigates the local structure of the moduli space of solutions to this system on a compact 6-manifold $X$, using a vector bundle $Q=(T^{1,0}X)^* \oplus {End}(E) \oplus T^{1,0}X$, where $E\to X$ is the classical gauge bundle arising in the system. We establish that the moduli space has an expected dimension of zero. We achieve this by studying the deformation complex associated to a differential operator $\bar{D}$, which emulates a holomorphic structure on $Q$, and demonstrating an isomorphism between the two cohomology groups which govern the infinitesimal deformations and obstructions in the deformation theory for the system. We also provide a Dolbeault-type theorem linking these cohomology groups to \v{C}ech cohomology, a result which might be of independent interest, as well as potentially valuable for future research.