Symplectic scalar curvature on supermanifolds

TL;DR

This paper explores the concept of symplectic scalar curvature on supermanifolds, introducing two families of odd super-Fedosov structures with differing curvature properties, including a specific example on a torus.

Contribution

It introduces two new classes of odd super-Fedosov structures on supermanifolds, one with trivial and one with non-trivial symplectic scalar curvature, expanding geometric understanding.

Findings

First family yields zero odd symplectic scalar curvature.

Second family exhibits non-trivial odd symplectic scalar curvature.

Explicit curvature calculation on the torus as a base manifold.

Abstract

We study the notion of symplectic scalar curvature on the supermanifold over an ordinary Fedosov manifold whose structural sheaf is that of differential forms. In this purely geometric context, we introduce two families of odd super-Fedosov structures, the first one is very general and uses a graded symmetric connection, leading to a vanishing odd symplectic scalar curvature, while the second one is based on a graded non-symmetric connection and has a non-trivial odd symplectic scalar curvature. As a simple example of the second case, we determine that curvature when the base Fedosov manifold is the torus.