Abstract
Heterotic string compactifications on integrable G2 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 G2 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.
Highlights
A heterotic G2 system is a quadruple ([Y, φ], [V, A], [T Y, θ], H ) where Y is a seven dimensional manifold with an integrable G2 structure φ, V is a bundle on Y with connection A, T Y is the tangent bundle of Y with connection θ, and H is a three form on Y determined uniquely by the G2 structure
This study is an extension of our work [49], where we determined the combined infinitesimal moduli space T M(Y,[V,A],[T Y,θ]) of heterotic G2 systems with H = 0, where Y is a G2 holonomy manifold
When combined with the instanton conditions on the bundles, we show that the constraints on the heterotic G2 system ([Y, φ], [V, A], [T Y, θ], H ) can be rephrased in terms of a nilpotency condition D 2 = 0 on the operator D acting on a bundle
Summary
A heterotic G2 system is a quadruple ([Y, φ], [V, A], [T Y, θ], H ) where Y is a seven dimensional manifold with an integrable G2 structure φ, V is a bundle on Y with connection A, T Y is the tangent bundle of Y with connection θ, and H is a three form on Y determined uniquely by the G2 structure. The G2 embeddings can be used to study flows of SU (3) structure manifolds [20,47,48] These results from physics and mathematics prompt and pave the way for our research on the combined infinitesimal moduli space T M of heterotic G2 systems ([Y, φ], [V, A], [T Y, θ], H ). This study is an extension of our work [49], where we determined the combined infinitesimal moduli space T M(Y,[V,A],[T Y,θ]) of heterotic G2 systems with H = 0, where Y is a G2 holonomy manifold. Our analysis complements the findings of [56], where methods of elliptic operator theory were used to show that the infinitesimal moduli space of heterotic G2 compactifications is finite dimensional when the G2 geometry is compact. Three appendices with useful formulas, curvature identities and a summary of heterotic supergravity complement the main discussion
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