Deconstructing Monopoles and Instantons

Abstract

We give a unifying description of the Dirac monopole on the 2-sphere $S^2$, of a graded monopole on a (2,2)-supersphere $S^{2,2}$ and of the BPST instanton on the 4-sphere $S^4$, by constructing a suitable global projector $p$ via equivariant maps. This projector determines the projective module of finite type of sections of the corresponding vector bundle. The canonical connection $\nabla = p \circ d$ is used to compute the topological charge which is found to be equal to -1 for the three cases. The transposed projector $q=p^t$ gives the value +1 for the charges; this showing that transposition of projectors, although an isomorphism in $K$-theory, is not the identity map. We also study the invariance under the action of suitable Lie groups.