On Calabi-Yau supermanifolds II

We study when Calabi-Yau supermanifolds M with one complex bosonic coordinate and two complex fermionic coordinates are super Ricci-flat, and find that if the bosonic manifold is compact, it must have constant scalar curvature. In [1], we found that super Ricci-flat Kähler manifolds with one fermionic dimension and an arbitrary number of bosonic dimensions exist above a bosonic manifold with a vanishing Ricci scalar. This paper explores super Calabi-Yau manifolds with one bosonic dimension and two fermionic dimensions. We find that the condition that the supermetric is super Ricciflat implies several interesting constraints that are familiar from other contexts, including the field equation of the WZW-model on AdS3. Locally, these constraints imply that the super Kähler potential has the form K(z, z̄, θ, θ̄) = K0(z, z̄) + √ K0(z, z̄),zz̄θ θ̄ + 1 4 (ln[K0(z, z̄),zz̄ ]),zz̄ (θ θ̄) , (1) where K0(z, z̄) is the Kähler potential of the bosonic manifold. We find the further constraint that the scalar curvature of the bosonic manifold is harmonic; on a complete compact space, this implies that the scalar curvature is constant. Consider the super Kähler potential K of the manifold M with 1 bosonic coordinate and 2 fermionic coordinates: K = f0 + if1θ 2 + if̄1θ̄ 2 + fij̄θ θ̄ + f2θ θ̄ (2) We use the notation θ = 1 2 ǫijθ θ, where ǫij = −ǫji. Since θiθj = θ̄ θ̄, the factor of i is needed to make K real. The supermetric of this manifold is