In previous works, an operator was developed for heterotic compactifications on ℝ2,1 × G2 and AdS3 × G2, which preserves N = 1 d = 3 supersymmetry and whose kernel is related to the moduli of the compactification. The operator is described in terms of non-physical spurious degrees of freedom, specifically, deformations of a connection on the tangent bundle. In this paper, we eliminate these spurious degrees of freedom by linking deformations of the spin connection to the moduli of the G2 manifold Y. This results in an operator \( \overset{\check{} }{D} \) that captures the physical moduli space of the G2 heterotic string theory. When Y = X × S1, with X an SU(3) manifold, we show \( \overset{\check{} }{D} \) produces results that align with existing literature. This allows us to propose a G2 moduli space metric. We check that this metric reduces to the SU(3) moduli metric constructed in the literature. We then define an adjoint operator \( {\overset{\check{} }{D}}^{\dagger } \) . We show the G2 moduli correspond to the intersection of the kernels of \( \overset{\check{} }{D} \) and \( {\overset{\check{} }{D}}^{\dagger } \) . These kernels reduce to the SU(3) F-terms and D-terms respectively on X × S1. This gives two non-trivial consistency checks of our proposed moduli space metric. Working perturbatively in α‵, we also demonstrate that the heterotic G2 moduli problem can be characterised in terms of a double extension of ordinary bundles, just like in the SU(3) case.
