Abstract
We describe the first order moduli space of heterotic string theory compactifications which preserve N =1 supersymmetry in four dimensions, that is, the infinitesimal parameter space of the Strominger system. We establish that if we promote a connection on T X to a field, the moduli space corresponds to deformations of a holomorphic structure $$ \overline{D} $$ on a bundle $$ \mathcal{Q} $$ . The bundle $$ \mathcal{Q} $$ is constructed as an extension by the cotangent bundle T ∗ X of the bundle E = End(V )⊕End(TX)⊕TX with an extension class $$ \mathcal{H} $$ which precisely enforces the anomaly cancelation condition. The deformations corresponding to the bundle E are simultaneous deformations of the holomorphic structures on the poly-stable bundles V and T X together with those of the complex structure of X. We discuss the fact that the “moduli” corresponding to End(T X) cannot be physical, but are however needed in our mathematical structure to be able to enforce the anomaly cancelation condition. In the appendix we comment on the choice of connection on T X which has caused some confusion in the community before. It has been shown by Ivanov and others that this connection should also satisfy the instanton equations, and we give another proof of this fact.
Highlights
In this paper, we study the heterotic string from the ten-dimensional supergravity point of view
We describe the first order moduli space of heterotic string theory compactifications which preserve N = 1 supersymmetry in four dimensions, that is, the infinitesimal parameter space of the Strominger system
As the stability of both V and T X is not spoiled by first order deformations of J which preserve the holomorphicity condition of both bundles, the theorem by Li and Yau guarantees that on the deformed heterotic compactification (X, V ) there is a connection on V
Summary
The heterotic compactification we are interested in consists on a pair (X, V ) where X is a six dimensional Riemannian spin manifold, together with a vector bundle V on X. This pair has the geometric properties given by. A six-dimensional compact space X with an SU(3)-structure given by a nowhere vanishing three-form Ψ, and a hermitian form ω satisfying the SU(3)-structure compatibility conditions. A connection ∇I on the tangent bundle with curvature RI which satisfies the instanton condition. As discussed in the appendix, this instanton connection is needed to ensure that supersymmetric solutions which satisfy the anomaly cancelation condition, solve the equations of motion.. We will defer the discussion of higher order α corrections to a future companion paper [41], but we comment briefly on the results of that paper in the appendix
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