Rank two quiver gauge theory, graded connections and noncommutative vortices

We consider equivariant dimensional reduction of Yang-Mills theory on K"ahlermanifolds of the form M times CP^1 times CP^1. This induces a rank two quivergauge theory on M which can be formulated as a Yang-Mills theory of gradedconnections on M. The reduction of the Yang-Mills equations on M times CP^1times CP^1 induces quiver gauge theory equations on M and quiver vortexequations in the BPS sector. When M is the noncommutative space R_theta^{2n}both BPS and non-BPS solutions are obtained, and interpreted as states ofD-branes. Using the graded connection formalism, we assign D0-brane charges inequivariant K-theory to the quiver vortex configurations. Some categoricalproperties of these quiver brane configurations are also described in terms ofthe corresponding quiver representations.Comment: 1+39 page