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

TL;DR

This paper explores instanton equations on Calabi-Yau and hyper-K"ahler cones, reducing them to matrix equations, and investigates boundary conditions, moduli spaces, and their relations to Nahm's equations and nilpotent structures.

Contribution

It introduces new boundary conditions for Hermitian Yang-Mills equations on Calabi-Yau cones and links generalized Nahm's equations to nilpotent pairs, extending Kronheimer's work.

Findings

New singular boundary conditions with known instanton solutions

Relation between generalized Nahm's equations and nilpotent pairs

Analysis of quaternionic instantons and HYM moduli spaces

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.