Projective Modules of Finite Type and Monopoles over $S^2$

TL;DR

This paper provides a unified framework for describing all vector bundles over the 2-sphere using projectors, and explores their topological charges and equivalences through explicit constructions.

Contribution

It introduces a method to construct global projectors for all vector bundles over $S^2$, linking algebraic and topological properties explicitly.

Findings

Constructed global projectors for all vector bundles over $S^2$

Computed topological charges using canonical connections

Showed transposition of projectors changes charge signs

Abstract

We give a unifying description of all inequivalent vector bundles over the 2-dimensional sphere $S^2$ by constructing suitable global projectors $p$ via equivariant maps. Each projector determines the projective module of finite type of sections of the corresponding complex rank 1 vector bundle over $S^2$. The canonical connection $\nabla = p \circ d$ is used to compute the topological charges. Transposed projectors gives opposite values for the charges, thus showing that transposition of projectors, although an isomorphism in K-theory, is not the identity map. Also, we construct the partial isometry yielding the equivalence between the tangent projector (which is trivial in K-theory) and the real form of the charge 2 projector.