Non-Abelian Cut Constructions and Hyperk\"ahler Modifications

TL;DR

This paper develops a general framework for non-Abelian cutting and modification constructions in hyperk"ahler geometry, extending symplectic cut techniques to more complex non-Abelian group actions with significant topological and metric effects.

Contribution

It introduces two novel non-Abelian hyperk"ahler modification methods, expanding the toolkit for geometric and topological changes in hyperk"ahler manifolds.

Findings

Two types of non-Abelian hyperk"ahler modifications introduced

Significant topological changes achieved through modifications

Distinct behavior of moment maps in non-Abelian hyperk"ahler actions

Abstract

We discuss a general framework for cutting constructions and reinterpret in this setting the work on non-Abelian symplectic cuts by Weitsman. We then introduce two analogous non-Abelian modification constructions for hyperk\"ahler manifolds: one modifies the topology significantly, the other gives metric deformations. We highlight ways in which the geometry of moment maps for non-Abelian hyperk\"ahler actions differs from the Abelian case and from the non-Abelian symplectic case.