Instantons on Calabi-Yau and hyper-K\"ahler cones

Abstract

The instanton equations on vector bundles over Calabi-Yau and hyper-K\"ahler 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-K\"ahler 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.