On Calabi-Yau supermanifolds
Martin Rocek, Neal Wadhwa
TL;DR
This paper demonstrates that Calabi-Yau supermanifolds with one fermionic dimension require the bosonic metric to have zero scalar curvature for a super Ricci-flat supermetric to exist, showing Yau's theorem does not extend to supermanifolds.
Contribution
It proves a necessary and sufficient condition for super Ricci-flat metrics on Calabi-Yau supermanifolds with one fermionic dimension, revealing limitations of Yau's theorem in supergeometry.
Findings
Super Ricci-flat supermetrics exist only if the bosonic metric has zero scalar curvature.
Yau's theorem does not hold for supermanifolds.
A characterization of super Ricci-flat metrics on Calabi-Yau supermanifolds.
Abstract
We prove that a Kahler 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.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Black Holes and Theoretical Physics