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

Abstract

We consider a complex vector bundle $${\mathcal{E}}$$ endowed with a connection $${\mathcal{A}}$$ over the eight-dimensional manifold $${\mathbb{R}^2\times G/H}$$ , where G/H = SU(3)/U(1) × 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 $${\mathcal{A}}$$ of the Spin(7)-instanton equations on $${\mathbb{R}^2\times G/H}$$ and general solutions of non-Abelian coupled vortex equations on $${\mathbb{R}^2}$$ . These vortices are BPS solitons in a d = 4 gauge theory obtained from $${\mathcal{N} =1}$$ 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 $${\mathbb{R}^2}$$ , are not holomorphic in the generic case. Finally, we introduce BPS vortex equations in $${\mathcal{N} =4}$$ super Yang–Mills theory and show that they have the same feature.