Symplectic geometries on supermanifolds

TL;DR

This paper extends symplectic geometry to supermanifolds, exploring even and odd cases, and introduces two distinct scalar symplectic structures in the odd case, enriching the geometric framework of supersymmetry.

Contribution

It provides a comprehensive extension of symplectic geometry to supermanifolds, including the classification of structures in even and odd cases, and introduces new scalar symplectic forms in the odd case.

Findings

In the even case, leads to even symplectic geometry and Fedosov supermanifolds.

In the odd case, identifies two different scalar symplectic structures: an odd 2-form and the antibracket.

Establishes the foundations for symplectic structures in supersymmetric geometric frameworks.

Abstract

Extension of symplectic geometry on manifolds to the supersymmetric case is considered. In the even case it leads to the even symplectic geometry (or, equivalently, to the geometry on supermanifolds endowed with a non-degenerate Poisson bracket) or to the geometry on an even Fedosov supermanifolds. It is proven that in the odd case there are two different scalar symplectic structures (namely, an odd closed differential 2-form and the antibracket) which can be used for construction of symplectic geometries on supermanifolds.