Holomorphic Bundles and the Moduli Space of N=1 Supersymmetric Heterotic Compactifications

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 $TX$ to a field, the moduli space corresponds to deformations of a holomorphic structure $\bar D$ on a bundle $\cal Q$. The bundle $\cal Q$ is constructed as an extension by the cotangent bundle $T^*X$ of the bundle $E= {\rm End}(V) \oplus {\rm End}(TX) \oplus TX$ with an extension class $\cal H$ 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 $TX$ together with those of the complex structure of $X$. We discuss the fact that the "moduli" corresponding to ${\rm End}(TX)$ 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 $TX$ 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.