J-holomorphic curves, moment maps, and invariants of Hamiltonian group actions

TL;DR

This paper develops invariants for Hamiltonian group actions on symplectic manifolds, extending Gromov-Witten invariants through PDE solutions involving the Cauchy-Riemann operator, curvature, and moment maps.

Contribution

It introduces an equivariant extension of Gromov-Witten invariants based on solutions to specific PDEs involving symplectic geometry tools.

Findings

Construction of new invariants for Hamiltonian actions

Extension of Gromov-Witten invariants to equivariant setting

Framework for PDE-based invariants involving moment maps

Abstract

We outline the construction of invariants of Hamiltonian group actions on symplectic manifolds. These invariants can be viewed as an equivariant version of Gromov-Witten invariants. They are derived from solutions of a PDE involving the Cauchy-Riemann operator, the curvature of a connection, and the moment map.