Fedosov supermanifolds: Basic properties and the difference in even and odd cases

TL;DR

This paper explores the properties of Fedosov supermanifolds with even and odd symplectic structures, highlighting differences in their curvature and Ricci tensors, and extending classical Fedosov geometry to supersymmetric contexts.

Contribution

It introduces a framework for Fedosov supermanifolds with symplectic structures, analyzing their curvature and tensor properties, and distinguishes between even and odd cases.

Findings

Odd Fedosov supermanifolds can have non-trivial scalar curvature.

Even Fedosov supermanifolds necessarily have zero scalar curvature.

The paper generalizes Fedosov geometry to supersymmetric supermanifolds.

Abstract

We study basic properties of supermanifolds endowed with an even (odd) symplectic structure and a connection respecting this symplectic structure. Such supermanifolds can be considered as generalization of Fedosov manifolds to the supersymmetric case. Choosing an apporpriate definition of inverse (second-rank) tensor fields on supermanifolds we consider the symmetry behavior of tensor fields as well as the properties of the symplectic curvature and of the Ricci tensor on even (odd) Fedosov supermanifolds. We show that for odd Fedosov supermanifolds the scalar curvature, in general, is non-trivial while for even Fedosov supermanifolds it necessarily vanishes.