Surjectivity for Hamiltonian Loop Group Spacees

TL;DR

This paper extends Kirwan's surjectivity theorem to Hamiltonian loop group spaces and quasi-Hamiltonian spaces, demonstrating that the inclusion of the zero level set induces a surjection in equivariant cohomology.

Contribution

It generalizes the surjectivity theorem to infinite-dimensional loop group spaces and quasi-Hamiltonian spaces using Morse theory techniques.

Findings

Surjectivity of the cohomology map for Hamiltonian loop group spaces.

Extension of Kirwan's theorem to infinite-dimensional settings.

Surjectivity result for quasi-Hamiltonian G-spaces.

Abstract

Let $G$ be a compact Lie group, and let $LG$ denote the corresponding loop group. Let $(X,\omega)$ be a weakly symplectic Banach manifold. Consider a Hamiltonian action of $LG$ on $(X,\omega)$, and assume that the moment map $\mu: X \to L\fg^*$ is proper. We consider the function $|\mu|^2: X \to \R$, and use a version of Morse theory to show that the inclusion map $j:\mu^{-1}(0)\to X$ induces a surjection $j^*:H_G^*(X) \to H_G^*(\mu^{-1}(0))$, in analogy with Kirwan's surjectivity theorem in the finite-dimensional case. We also prove a version of this surjectivity theorem for quasi-Hamiltonian $G$-spaces.