Abstract
Heterotic string compactifications on integrable G 2 structure manifolds Y with instanton bundles \({(V,A), (TY,\tilde{\theta})}\) yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative \({\mathcal{D}}\) and show that it is equivalent to a heterotic G 2 system encoding the geometry of the heterotic string compactifications. This operator \({\mathcal{D}}\) acts on a bundle \({\mathcal{Q}=T^*Y \oplus {\rm End}(V) \oplus {\rm End}(TY)}\) and satisfies a nilpotency condition \({\check{{\mathcal{D}}}^2=0}\) , for an appropriate projection of \({\mathcal D}\). Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group \({\check H^1_{\check{{\mathcal{D}}}}(\mathcal{Q})}\). We comment on the similarities and differences of our result with Atiyah’s well-known analysis of deformations of holomorphic vector bundles over complex manifolds. Our analysis leads to results that are of relevance to all orders in the \({\alpha'}\) expansion.
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de la Ossa, X., Larfors, M. & Svanes, E.E. The Infinitesimal Moduli Space of Heterotic G 2 Systems. Commun. Math. Phys. 360, 727–775 (2018). https://doi.org/10.1007/s00220-017-3013-8
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DOI: https://doi.org/10.1007/s00220-017-3013-8