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

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 $S^{2,2}$.