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

We construct explicit BPS and non-BPS solutions of the U(2k) Yang-Mills equations on the noncommutative space R θ ×S2 with finite energy and topological charge. By twisting with a Dirac multi-monopole bundle over S, we reduce the Donaldson-Uhlenbeck-Yau equations on R θ ×S2 to vortex-type equations for a pair of U(k) gauge fields and a bi-fundamental scalar field on R θ . In the SO(3)-invariant case the vortices on R θ determine multi-instantons on R θ ×S2. 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 D0branes on R θ into spherical D2-branes on R θ ×S2. In the generic case with broken rotational symmetry, we argue that the D0-brane charges on R θ ×S2 provide a physical interpretation of the Adams operations in K-theory.