Deconstructing Monopoles and Instantons
Abstract
Description
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.
latex, 23 pages (one additional page). Additional material at the end of Section 2. and a reference added. Several minor additions throughout the paper
latex, 23 pages (one additional page). Additional material at the end of Section 2. and a reference added. Several minor additions throughout the paper