Yang–Mills connections over manifolds with Grassmann structure

Abstract

Let M be a manifold with Grassmann structure, i.e., with an isomorphism of the cotangent bundle T*M≅E⊗H with the tensor product of two vector bundles E and H. We define the notion of a half-flat connection ∇W in a vector bundle W→M as a connection whose curvature F∈S2E⊗Λ2H⊗End W ⊂ Λ2T*M⊗End W. Under appropriate assumptions, for example, when the Grassmann structure is associated with a quaternionic Kähler structure on M, half-flatness implies the Yang–Mills equations. Inspired by the harmonic space approach, we develop a local construction of (holomorphic) half-flat connections ∇W over a complex manifold with (holomorphic) Grassmann structure equipped with a suitable linear connection. Any such connection ∇W can be obtained from a prepotential by solving a system of linear first order ODEs. The construction can be applied, for instance, to the complexification of hyper-Kähler manifolds or more generally to hyper-Kähler manifolds with admissible torsion and to their higher-spin analogs. It yields solutions of the Yang–Mills equations.