Projective Modules of Finite Type over the Supersphere $S^{2,2}$

TL;DR

This paper constructs all inequivalent vector bundles over the supersphere $S^{2,2}$ using projectors, computes their topological charges via Chern numbers, and explores their K-theory, extending concepts of monopoles to a supergeometric setting.

Contribution

It provides a complete classification of vector bundles over the supersphere $S^{2,2}$ using global projectors and analyzes their topological properties.

Findings

Constructed all inequivalent vector bundles over $S^{2,2}$.

Computed Chern numbers for these bundles using Berezin integral.

Identified supertransposed projectors as having opposite topological charges.

Abstract

In the spirit of noncommutative geometry we construct all inequivalent vector bundles over the $(2,2)$-dimensional supersphere $S^{2,2}$ by means of global projectors $p$ via equivariant maps. Each projector determines the projective module of finite type of sections of the corresponding `rank 1' supervector bundle over $S^{2,2}$. The canonical connection $\nabla = p \circ d$ is used to compute the Chern numbers by means of the Berezin integral on $S^{2,2}$. The associated connection 1-forms are graded extensions of monopoles with not trivial topological charge. Supertransposed projectors gives opposite values for the charges. We also comment on the $K$-theory of $S^{2,2}$.