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Almost Parity Structure, Connections and Vielbeins in BV Geometry

K. Bering
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Abstract

We observe that an anti-symplectic manifold locally always admits a parity structure. The parity structure can be viewed as a complex-like structure on the manifold. This induces an odd metric and its Levi-Civita connection, and thereby a new notion of an odd Kaehler geometry. Oversimplified, just to capture the idea, the bosonic variables are ``holomorphic'', while the fermionic variables are ``anti-holomorphic''. We find that an odd Kaehler manifold in this new ``complex'' sense has a nilpotent odd Laplacian iff it is Ricci-form-flat. The local cohomology of the odd Laplacian is derived. An odd Calabi-Yau manifold has locally a canonical volume form. We suggest that an odd Calabi-Yau manifold is the natural geometric notion to appear in covariant BV-quantization. Finally, we give a vielbein formulation of anti-symplectic manifolds.

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K. Bering. 1998-02-06. Almost Parity Structure, Connections and Vielbeins in BV Geometry. https://arxiv.org/abs/physics/9711010

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