# Almost Parity Structure, Connections and Vielbeins in BV Geometry

Source: [Almost Parity Structure, Connections and Vielbeins in BV Geometry](<https://www.sota2.com/research/paper/68fa21f87a2b558e29339c8c>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T02:23:59.896Z

Authors: K. Bering

Published: 1997-11-12T00:29:07.000Z

## 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.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/physics/9711010>)

[PDF](<https://arxiv.org/pdf/physics/9711010.pdf>)