A Heterotic Hermitian--Yang--Mills Equivalence

We consider N=1, d=4 vacua of heterotic theories in the large radius limit in which alpha' << 1. We construct a real differential operator D= D+D̄ on an extension bundle (Q, D) with underlying topology Q=(T1,0X)^* ⊕ End \, E ⊕ T1,0 X whose curvature is holomorphic and Hermitian-Yang-Mills with respect to the complex structure and metric on the underlying non-Kahler complex 3-fold X if and only if the heterotic supersymmetry equations and Bianchi identity are satisfied. This is suggestive of an analogue of the Donaldson--Uhlenbeck--Yau correspondence for heterotic vacua of this type.