On the symplectic structure of instanton moduli spaces

TL;DR

This paper explores the complex symplectic structure of instanton moduli spaces via quiver varieties, identifying Darboux coordinates, Hamiltonians, and symplectomorphism actions, extending results from Calogero-Moser spaces.

Contribution

It introduces a detailed analysis of the symplectic structure of instanton moduli spaces, including global coordinates and symmetry group actions, bridging to Calogero-Moser spaces.

Findings

Identification of Darboux coordinates for instanton moduli spaces

Construction of quadratic Hamiltonians on classical phase spaces

Transitive action of non-commutative symplectomorphisms

Abstract

We study the complex symplectic structure of the quiver varieties corresponding to the moduli spaces of SU(2) instantons on both commutative and non-commutative R^4. We identify global Darboux coordinates and quadratic Hamiltonians on classical phase spaces for which these quiver varieties are natural completions. We also show that the group of non-commutative symplectomorphisms of the corresponding path algebra acts transitively on the moduli spaces of non-commutative instantons. This paper should be viewed as a step towards extending known results for Calogero-Moser spaces to the instanton moduli spaces.