On Calabi-Yau supermanifolds

We prove that a Kähler supermetric on a supermanifold with one complex fermionic dimension admits a super Ricci-flat supermetric if and only if the bosonic metric has vanishing scalar curvature. As a corollary, it follows that Yau’s theorem does not hold for supermanifolds. Calabi[1] proposed that if a Kähler manifold has vanishing first Chern class, that is, the Ricci-form obeys Rij̄(g) = ∂iv̄j − ∂̄jvi for a globally defined 1-form v, or, equivalently, a complex n-dimensional Kähler manifold has a globally defined holomorphic top form Ωi1...in , then there exists a unique metric g which is a smooth deformation of g and obeys Rij̄(g ) = 0. Yau[2] proved this theorem for ordinary manifolds. Recently, there has been a lot of interest in Calabi-Yau supermanifolds [3]; though these papers use only the topological properties of such spaces, it is interesting to ask whether they also admit Ricci-flat supermetrics. This paper studies the generalization of Calabi’s conjecture to supermanifolds with one complex fermionic dimension. We find that such a Kähler supermanifold admits a Ricci-flat supermetric if and only if the bosonic metric has vanishing scalar curvature. For a given scalar-flat bosonic Kähler metric with Kähler potential KBose, the super-extension is unique, and has the super Kähler potential: K(z, z̄ , θ, θ̄) = KBose(z , z̄) + det ( ∂ ∂zi∂z̄j KBose ) θθ̄ . (1) As complex projective spaces do not admit scalar-flat metrics, but do admit super CalabiYau extensions with one fermionic dimension, it follows that Yau’s theorem does not hold for supermanifolds. A supermanifold is a generalization of a usual manifold with fermionic as well as bosonic coordinates. The bosonic coordinates are ordinary numbers, whereas the fermionic coordinates are grassmann numbers. Grassmann numbers are odd elements of a grassmann algebra and anticommute: θθ = −θθ and θθ = 0. On bosonic Kähler manifolds, the Ricci tensor Rij̄ = (ln det(g)),ij̄ . (2) For this to vanish, ln det(g)) (locally) must be the real part of a holomorphic function, and hence det(g)) = |f(z)| for some holomorphic f(z). This can always be absorbed by a holomorphic coordinate transformation, and hence a Kähler manifold is Ricci-flat if its Kähler potential K obeys the Monge-Ampère equation det(g) ≡ det(K,ij̄) = 1 . (3) On supermanifolds, because elements of g contain grassmann numbers, the determinant is not well defined and a new definition of the determinant is needed. For any nondegenerate supermatrix g = ( A B C D ) , (4) where A and D are bosonic and B and C are fermionic, sdet(g) ≡ det(A) det(D − CAB) = det(A− BDC) det(D) . (5) More rigorous and technical definitions can be found in the literature, see, e.g., [4], but this simple treatment suffices for our results.