Localization for nonabelian group actions

TL;DR

This paper derives a residue formula to evaluate classes on the symplectic reduction of a compact Hamiltonian K-manifold, extending localization techniques to nonabelian group actions.

Contribution

It provides a new residue formula for integrating classes over nonabelian symplectic quotients, generalizing abelian localization methods.

Findings

Derived a residue formula for nonabelian group actions

Connected equivariant cohomology to reduced space cohomology

Extended localization techniques to nonabelian symplectic reductions

Abstract

Suppose $X$ is a compact symplectic manifold acted on by a compact Lie group $K$ (which may be nonabelian) in a Hamiltonian fashion, with moment map $\mu: X \to {\rm Lie}(K)^*$ and Marsden-Weinstein reduction $\xred = \mu^{-1}(0)/K$. There is then a natural surjective map $\kappa_0$ from the equivariant cohomology $H^*_K(X) $ of $X$ to the cohomology $H^*(\xred)$. In this paper we prove a formula (Theorem 8.1, the residue formula) for the evaluation on the fundamental class of $\xred$ of any $\eta_0 \in H^*(\xred)$ whose degree is the dimension of $\xred$, provided that $0$ is a regular value of the moment map $\mu$ on $X$. This formula is given in terms of any class $\eta \in H^*_K(X)$ for which $\kappa_0(\eta ) = \eta_0$, and involves the restriction of $\eta$ to $K$-orbits $KF$ of components $F \subset X$ of the fixed point set of a chosen maximal torus $T \subset K$. Since $\kappa_0$ is