TL;DR
This paper investigates Ricci flatness conditions on supermanifolds, revealing that certain properties hold only for fermionic complex dimension one and proposing a conjecture extending Calabi-Yau theorem to higher dimensions.
Contribution
It proves that the Ricci flatness phenomenon is exclusive to fermionic complex dimension one and conjectures the extension of Calabi-Yau theorem to higher dimensions.
Findings
Ricci flatness implies scalar-flat bosonic submanifold for fermionic dimension one
The Ricci flatness phenomenon does not generalize to higher fermionic dimensions
Conjecture that Calabi-Yau theorem extends to supermanifolds with larger fermionic dimension
Abstract
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.
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