# The Infinitesimal Moduli Space of Heterotic G$_{2}$ Systems

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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Commun.Math.Phys., 2018, 360 (2), pp.727-775. 〈10.1007/s00220-017-3013-8〉
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https://hal.archives-ouvertes.fr/hal-01792508
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Soumis le : mardi 15 mai 2018 - 15:13:33
Dernière modification le : lundi 13 août 2018 - 19:25:50

### Citation

Xenia De La Ossa, Magdalena Larfors, Eirik Svanes. The Infinitesimal Moduli Space of Heterotic G$_{2}$ Systems. Commun.Math.Phys., 2018, 360 (2), pp.727-775. 〈10.1007/s00220-017-3013-8〉. 〈hal-01792508〉

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