Note that 6-manifolds with SU(3) structure (i.e. SU(3) holonomy of the spin connection with torsion proportional to the $H$-field) can be described in terms of conditions on torsion classes of these manifolds [@CS]. Due to the inclusion of the $H$-field in the geometry of the internal manifold as torsion, we have to deal with non-Kähler and in some cases non-complex manifolds.

Most research in flux compactifications was done in type II string theories (see e.g. [@Grana]-[@BKLS] and [@group0]-[@group2] for more recent results). However, fluxes in heterotic string theory, which play a prominent role in stringy model building, have been considered as well (see e.g.  [@FL; @BLP; @Zoupanos], [@LM]-[@papa] and references therein). Historically, heterotic flux compactifications have been known for quite some time, starting with the works [@SHW] in the mid-1980’s. In heterotic string compactifications one has the freedom to choose a gauge bundle since the simple embedding of the spin connection into the gauge connection is ruled out for compactifications with $dH\ne 0$. For the torsionful background, the allowed gauge bundle is restricted by the Hermitian-Yang-Mills equations [@Don; @UY] and by the Bianchi identity for the $H$-field (anomaly cancellation). The existence of such vector bundles on some non-Kähler [*complex*]{} 3-folds, 