Non-Abelian Vortices, Super-Yang-Mills Theory and Spin(7)-Instantons
Abstract
We consider a complex vector bundle E endowed with a connection A over the eight-dimensional manifold R2 x G/H, where G/H = SU(3)/U(1)xU(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 R2 x G/H and general solutions of non-Abelian coupled vortex equations on R2. 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 R2, 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.