Super Symplectic Geometry and Prequantization

TL;DR

This paper explores super symplectic geometry and prequantization, revealing that non homogeneous forms alter the structure of the Poisson algebra and affect the equivalence of prequantization methods.

Contribution

It introduces the theory of non homogeneous super symplectic forms and analyzes their impact on prequantization, Poisson algebra, and related geometric structures.

Findings

Poisson algebra becomes a subset of functions with super vector space values

No significant impact on coadjoint orbits, momentum maps, and central extensions

Prequantization methods diverge for non even symplectic forms

Abstract

We review the prequantization procedure in the context of super symplectic manifolds with a symplectic form which is not necessarily homogeneous. In developing the theory of non homogeneous symplectic forms, there is one surprising result: the Poisson algebra no longer is the set of smooth functions on the manifold, but a subset of functions with values in a super vector space of dimension 1|1. We show that this has no notable consequences for results concerning coadjoint orbits, momentum maps, and central extensions. Another surprising result is that prequantization in terms of complex line bundles and prequantization in terms of principal circle bundles no longer are equivalent if the symplectic form is not even.