Non-Abelian Vortices, Super Yang-Mills Theory and Spin(7)-Instantons

We consider a complex vector bundle epsilon endowed with a connection A over the eight-dimensional manifold R-2 x G/H, where G/H = SU(3)/U(1) x U(1) is a homogeneous space provided with a never-integrable almost-complex structure and a family of SU(3)-structures. We establish an equivalence between G-invariant solutions A of the Spin(7)-instanton equations on R-2 x G/H and general solutions of non-Abelian coupled vortex equations on R-2. These vortices are BPS solitons in a d = 4 gauge theory obtained from N = 1 supersymmetric Yang-Mills theory in ten dimensions compactified on the coset space G/H with an SU(3)-structure. The novelty of the obtained vortex equations lies in the fact that Higgs fields, defining morphisms of vector bundles over R-2, are not holomorphic in the generic case. Finally, we introduce BPS vortex equations in N = 4 super Yang-Mills theory and show that they have the same feature.