Algebraic classification of \({\mathbb{C}}P^ n\) instanton solutions
Instanton solutions of \({\mathbb{C}}P^ n\) models defined on orientable two- dimensional Riemann surfaces \(S_ g\) are the topic of this article (g is the topologically-invariant number of handles on \(S_ g)\). Instanton solutions are defined to be harmonic maps \(\phi:S_ g\to {\mathbb{C}}P^ n\) with finite action. The harmonicity condition means that the map \(\phi\) is a local extremum of the action, and hence a solution of the equations of motion. Harmonicity therefore replaces the latter, enabling a global analysis and characterization of instantons in terms of the properties of harmonic maps to be given. It follows from standard results in algebraic geometry that instantons are rational maps, as one knows for the familiar manifolds \(E^ 2\), \(S^ 2\). Essentially, the present work is an effort toward systematic generalization of known results. Following some analysis of a general nature, the map \(\phi:S^2\to {\mathbb{C}}P^ n\) is used as an explicit illustration. The number of parameters for this \(g=0\) case is obtained. For more general manifolds, constraints on the number of instanton parameters are given. A subsequent and more detailed version of this paper was scheduled to appear.
orientable two- dimensional Riemann surfaces
scientific article; zbMATH DE number 3845733