Instantons on Calabi-Yau cones

TL;DR

This paper reduces the Hermitian Yang-Mills equations on Calabi-Yau cones to Nahm-type matrix equations and demonstrates that, under certain conditions, these can be simplified to holomorphicity conditions, providing new insights into instanton solutions.

Contribution

It generalizes Donaldson and Kronheimer's methods to analyze instantons on Calabi-Yau cones, reducing complex equations to simpler holomorphic conditions.

Findings

Reduction of Hermitian Yang-Mills equations to Nahm-type equations

Demonstration that instanton equations can be simplified to holomorphicity conditions

Unique gauge transformation leading to simplified equations

Abstract

The Hermitian Yang-Mills equations on certain vector bundles over Calabi-Yau cones can be reduced to a set of matrix equations; in fact, these are Nahm-type equations. The latter can be analysed further by generalising arguments of Donaldson and Kronheimer used in the study of the original Nahm equations. Starting from certain equivariant connections, we show that the full set of instanton equations reduce, with a unique gauge transformation, to the holomorphicity condition alone.