We consider \(N=1\) , \(d=4\) vacua of heterotic theories in the large radius limit in which \({{\alpha }^{\backprime }\,}\ll 1\) . We construct a real differential operator \(\mathcal {D}= D+\bar{D}\) on an extension bundle \((Q, \mathcal {D})\) with underlying topology \(Q=(T^{1,0}X)^* \oplus \textrm{End} \, E \oplus T^{1,0} X\) whose curvature is holomorphic and Hermitian–Yang–Mills with respect to the complex structure and metric on the underlying non-Kähler 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.
