On Non-Abelian Symplectic Cutting

TL;DR

This paper explores symplectic cutting for non-Abelian group actions, providing a global quotient description via degeneration techniques and connecting it to algebro-geometric interpretations.

Contribution

It introduces a degeneration approach using the Vinberg monoid to describe symplectic cuts for non-Abelian groups and relates it to moduli spaces of framed bundles.

Findings

Global quotient description of symplectic cuts for non-Abelian groups

Connection between symplectic cutting and algebro-geometric moduli spaces

Identification of the universal cut with a moduli space of framed bundles

Abstract

We discuss symplectic cutting for Hamiltonian actions of non-Abelian compact groups. By using a degeneration based on the Vinberg monoid we give, in good cases, a global quotient description of a surgery construction introduced by Woodward and Meinrenken, and show it can be interpreted in algebro-geometric terms. A key ingredient is the `universal cut' of the cotangent bundle of the group itself, which is identified with a moduli space of framed bundles on chains of projective lines recently introduced by the authors.