Localization for nonabelian group actions

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 $\text{μ: X →}\text{Lie}\text{(K)}{}^{\mathrm{\ast }}$ and Marsden-Weinstein reduction $\text{M}{}_{\mathrm{X}}\text{=}\text{μ}{}^{\mathrm{-1}}\text{(0)}\text{K}$. There is then a natural surjective map κ₀ from the equivariant cohomology $\text{H}{}^{\mathrm{\ast }}{}_{\mathrm{K}}\text{(X)}$ of X to the cohomology $\text{H}{}^{\mathrm{\ast }}\text{(}\text{M}{}_{\mathrm{X}}\text{)}$. In this paper we prove a formula (Theorem 8.1, the residue formula) for the evaluation on the fundamental class of $\text{M}{}_{\mathrm{X}}$ of any $\text{η}{}_0\text{ϵ H}{}^{\mathrm{\ast }}\text{(}\text{M}{}_{\mathrm{X}}\text{)}$ whose degree is the dimension of $\text{M}{}_{\mathrm{X}}$, provided that 0 is a regular value of the moment map μ on X. This formula is given in terms of any class $\text{ηϵ H}{}^{\mathrm{\ast }}{}_{\mathrm{K}}\text{(X)}$ for which κ₀(η) = η₀, and involves the restriction of η to K-orbits KF of components F ⊂X of the fixed point set of a chosen maximal torus T ⊂K. Since κ₀ is surjective, in principle the residue formula enables one to determine generators and relations for the cohomology ring $\text{H}{}^{\mathrm{\ast }}\text{(}\text{M}{}_{\mathrm{X}}\text{)}$, in terms of generators and relations for $\text{H}{}^{\mathrm{\ast }}{}_{\mathrm{K}}\text{(X)}$. There are two main ingredients in the proof of our formula: one is the localization theorem [3,7] for equivariant cohomology of manifolds acted on by compact abelian groups, while the other is the equivariant normal form for the symplectic form near the zero locus of the moment map.

We also make use of the techniques appearing in our proof of the residue formula to give a new proof of the nonabelian localization formula of Witten ([35, Section 2]) for Hamiltonian actions of compact groups K on symplectic manifolds X; this theorem expresses $\text{η}{}_0\text{[}\text{M}{}_{\mathrm{X}}\text{]}$ in terms of certain integrals over X.