Instantons on Calabi-Yau and hyper-Kähler cones

Abstract

The instanton equations on vector bundles over Calabi-Yau and hyper-Kähler cones can be reduced to matrix equations resembling Nahm's equations. We complement the discussion of Hermitian Yang-Mills (HYM) equations on Calabi-Yau cones, based on regular semi-simple elements, by a new set of (singular) boundary conditions which have a known instanton solution in one direction. This approach extends the classic results of Kronheimer by probing a relation between generalised Nahm's equations and nilpotent pairs/tuples. Moreover, we consider quaternionic instantons on hyper-Kähler cones over generic 3-Sasakian manifolds and study the HYM moduli spaces arising in this set-up, using the fact that their analysis can be traced back to the intersection of three Hermitian Yang-Mills conditions.

Correction to: Affine connections on 3-Sasakian homogeneous manifolds

“…Moreover, for 7-dimensional Riemannian manifolds, the existence of three Killing spinors is equivalent to the existence of a 3-Sasakian structure, [28]. Bearing in mind that cones over 3-Sasakian manifolds produce Calabi-Yau manifolds, recent developments on 3-Sasakian geometry also include the Yang-Mills equations on cones over 3-Sasakian manifolds ( [29] and references therein). Finally, in the study of the control system of a n-dimensional Riemannian manifold M rolling on the sphere S n , without twisting or slipping, and under certain assumption, the manifold M can be endowed with a 3-Sasakian structure, [17].…”