Seiberg–Witten monopole equations on noncommutative R4

Abstract

It is well known that, due to vanishing theorems, there are no nontrivial finite action solutions to the Abelian Seiberg–Witten (SW) monopole equations on Euclidean four-dimensional space R4. We show that this is no longer true for the noncommutative version of these equations, i.e., on a noncommutative deformation Rθ4 of R4 there exist smooth solutions to the SW equations having nonzero topological charge. We introduce action functionals for the noncommutative SW equations and construct explicit regular solutions. All our solutions have finite energy. We also suggest a possible interpretation of the obtained solutions as codimension four vortex-like solitons representing D(p−4)- and D(p−4)¯-branes in a Dp-Dp¯ brane system in type II superstring theory.