Sasakian quiver gauge theory on the Aloff–Wallach space X1,1

Abstract

We consider the SU(3)-equivariant dimensional reduction of gauge theories on spaces of the form Md×X1,1 with d-dimensional Riemannian manifold Md and the Aloff–Wallach space X1,1=SU(3)/U(1) endowed with its Sasaki–Einstein structure. The condition of SU(3)-equivariance of vector bundles, which has already occurred in the studies of Spin(7)-instantons on cones over Aloff–Wallach spaces, is interpreted in terms of quiver diagrams, and we construct the corresponding quiver bundles, using (parts of) the weight diagram of SU(3). We consider three examples thereof explicitly and then compare the results with the quiver gauge theory on Q3=SU(3)/(U(1)×U(1)), the leaf space underlying the Sasaki–Einstein manifold X1,1. Moreover, we study instanton solutions on the metric cone C(X1,1) by evaluating the Hermitian Yang–Mills equation. We briefly discuss some features of the moduli space thereof, following the main ideas of a treatment of Hermitian Yang–Mills instantons on cones over generic Sasaki–Einstein manifolds in the literature.