Noncommutative Instantons in Higher Dimensions, Vortices and Topological K-Cycles

TL;DR

This paper constructs explicit noncommutative instanton and vortex solutions on higher-dimensional spaces, linking them to topological K-theory concepts like Bott periodicity and D-brane dynamics.

Contribution

It introduces new explicit solutions of Yang-Mills equations on noncommutative spaces and connects these to topological K-homology and D-brane physics.

Findings

Solutions realize Bott periodicity and vector bundle modification.

D0-branes can blow up into spherical D2-branes on noncommutative spaces.

D0-brane charges relate to Adams operations in K-theory.

Abstract

We construct explicit BPS and non-BPS solutions of the U(2k) Yang-Mills equations on the noncommutative space R^{2n}_\theta x S^2 with finite energy and topological charge. By twisting with a Dirac multi-monopole bundle over S^2, we reduce the Donaldson-Uhlenbeck-Yau equations on R^{2n}_\theta x S^2 to vortex-type equations for a pair of U(k) gauge fields and a bi-fundamental scalar field on R^{2n}_\theta. In the SO(3)-invariant case the vortices on R^{2n}_\theta determine multi-instantons on R^{2n}_\theta x S^2. We show that these solutions give natural physical realizations of Bott periodicity and vector bundle modification in topological K-homology, and can be interpreted as a blowing-up of D0-branes on R^{2n}_\theta into spherical D2-branes on R^{2n}_\theta x S^2. In the generic case with broken rotational symmetry, we argue that the D0-brane charges on R^{2n}_\theta x S^2 provide a physical interpretation of the Adams operations in K-theory.