×

Localization for nonabelian group actions. (English) Zbl 0833.55009

Close ties between symplectic geometry and equivariant cohomology began with the work of Atiyah and Bott in the early eighties. Since then the area attracted the interest of many analysts and geometers, with the influx also of ideas from theoretical physics, specially gauge theory.
From the summary: “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 \(\mu: X\to \text{Lie} (K)^*\) and Marsden- Weinstein reduction \({\mathcal M}_X= \mu^{-1} (0)/K\). There is then a natural surjective map \(h_0\) from the equivariant cohomology \(H^*_K (X)\) of \(X\) to the cohomology \(H^* ({\mathcal M}_X)\). In this paper we prove a formula (Theorem 8.1, the residue formula) for the evaluation on the fundamental class of \({\mathcal M}_X\) of any \(\eta_0\in H^* ({\mathcal M}_X)\) whose degree is the dimension of \({\mathcal M}_X\), provided 0 is a regular value of the moment map \(\mu\) on \(X\dots\). Since \(h_0\) is surjective, in principle the residue formula enables one to determine generators and relations for the cohomology ring \(H^* ({\mathcal M}_X)\), in terms of generators and relations for \(H^*_K (X)\dots\). We also make use of the techniques in our proof to give a new proof of the nonabelian localization formula of Witten for Hamiltonian actions of compact groups \(K\) on symplectic manifolds \(X\)”.

MSC:

55N91 Equivariant homology and cohomology in algebraic topology
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
57S25 Groups acting on specific manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv