Symplectic Connections on Supermanifolds: Existence and Non-Uniqueness

TL;DR

This paper proves that on any symplectic supermanifold, there exists an infinite-dimensional space of symmetric connections compatible with the symplectic form, highlighting non-uniqueness and abundance of such structures.

Contribution

It establishes the existence of infinitely many compatible symmetric connections on symplectic supermanifolds, demonstrating non-uniqueness in this geometric setting.

Findings

Existence of infinite-dimensional affine space of symmetric connections

Compatibility of these connections with the symplectic form

Non-uniqueness of symplectic connections on supermanifolds

Abstract

We show, in this note, that on any symplectic supermanifold, even or odd, there exist an infinite dimensional affine space of symmetric connections, compatible to the symplectic form.