On Ricci flat supermanifolds

We study the Ricci flatness condition on generic supermanifolds. It has been found recently that when the fermionic complex dimension of the supermanifold is one the vanishing of the super-Ricci curvature implies the bosonic submanifold has vanishing scalar curvature. We prove that this phenomena is only restricted to fermionic complex dimension one. Further we conjecture that for complex fermionic dimension larger than one the Calabi-Yau theorem holds for supermanifolds. Calabi-Yau compactification has been one of the most important cornerstones of superstring phenomenology. It is a supersymmetric vacuum solution to string theory in the absence of RR and NS-NS fluxes. Much progress has been made in studying sigma models, topological models and branes on Calabi-Yau backgrounds. The extension to Calabi-Yau supermanifold had been attempted several years ago [1, 2], and received much attention recently after Witten’s proposal that the perturbative amplitudes of the N = 4 super Yang-Mills theory can be recovered from open string theory on the Calabi-Yau supermanifold CP [3]. It belongs to a class of supermanifolds which can be obtained starting from a certain bosonic vector bundle over a Kähler manifold and then fermionizing the bundle direction. The global holomorphic top form exists as long as the base manifold and the vector bundle have the same canonical line bundle. It is reasonable to compactify the string theory on supermanifolds and look for conformal backgrounds. By the famous Calabi-Yau theorem, for given complex structure and Kähler class on a Kähler manifold, there exists a unique Ricci flat metric if and only if the first Chern class of the manifold vanishes, or there is a globally defined holomorphic top form on the manifold. Since the worldsheet sigma model is conformal invariant only when the target space is Ricci flat, it follows that the above-mentioned class of supermanifolds are all valid perturbative string theory backgrounds. It is then a surprise that Rǒcek and Wadhwa [4] found a counterexample to the CalabiYau theorem when the supermanifolds are constructed by fermionizing a line bundle over the base manifold. They proved that in this case, the super-Ricci flatness actually requires more than just the vanishing of the first Chern class: it also requires the bosonic base manifold to have vanishing scalar curvature. The novelty of the supermanifold compared to the bosonic manifold, regarding the Ricci flatness, is that in addition to the vanishing of first Chern class as an integrability condition, there are local constraints from the fermionic expansion of the curvature, as we will analyze later. We will retain the name Calabi-Yau manifolds for Kähler manifolds with vanishing first Chern class, or equivalently with a global holomorphic top form, in the case of supermanifolds. So by the result of [4], CP is a super Calabi-Yau manifold, as it has a global holomorphic (3, 1) form, but it is not super Ricci-flat as the base manifold CP has nonvanishing scalar curvature. A natural question is whether this counterexample to the Calabi-Yau theorem is merely an exception restricted to fermionic complex dimension one, or more general valid for higher fermionic dimension. We will show in this paper that for the Ricci-flat metric on supermanifold to imply that the bosonic manifold has vanishing scalar curvature, the condition of the complex fermionic dimension being one is not only sufficient but also necessary. An intuitive