Local Descriptions of the Heterotic SU(3) Moduli Space

The heterotic \(\textrm{SU}(3)\) 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 \(Q=(T^{1,0}X)^* \oplus {{\textrm{End}}}(E) \oplus T^{1,0}X\), where \(E\rightarrow X\) 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 Čech cohomology, a result which might be of independent interest, as well as potentially valuable for future research.