Equivariant localization and stationary phase.

*(English)*Zbl 1033.57016
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6–15, 2002. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-4-1/pbk). 88-124 (2003).

The author discusses Hamiltonian actions on symplectic manifolds and gives a self-contained introduction to the Cartan model of equivariant cohomology. He relates the results to the Cartan theorem asserting that the \(G\)-equivariant cohomology algebra \(H^*_G(M)\) is isomorphic to the de Rham cohomology with complex coefficients of the orbit manifold \(M/G\), where \(G\) is a compact connected Lie group acting smoothly and freely on a smooth manifold \(M\). Then the author proves the major result of the paper, the equivariant localization theorem about computing \(\int_M\alpha(\xi)\) for any \(G\)-equivariantly closed differential form \(\alpha\) on \(M\) and any nondegenerate element \(\xi\in{\mathfrak g}\) for which the associated vector field \(\xi^{\#}\) has only isolated zeros, where \({\mathfrak g}\) is the Lie algebra of a compact Lie group \(G\) acting smoothly on a compact oriented manifold \(M\) of dimension \(2k\). As an application of the theorem, the author derives the generalized Duistermaat-Heckman theorem about computing \(\int_M e^{i\mu(\xi)}\nu_\omega\) for any compact symplectic manifold \((M,\omega)\) of dimension \(2k\) with a Hamiltonian action of \(G\) and corresponding symplectic moments given by \(\mu:{\mathfrak g}\to C^\infty(M)\), where \(M\) is oriented with the Liouville form \(\nu_\omega= {1\over k!} \omega^k\).

For the entire collection see [Zbl 1008.00022].

For the entire collection see [Zbl 1008.00022].

Reviewer: Krzysztof Pawałowski (Poznań)

##### MSC:

57R91 | Equivariant algebraic topology of manifolds |

53D05 | Symplectic manifolds, general |

53D35 | Global theory of symplectic and contact manifolds |

55N91 | Equivariant homology and cohomology in algebraic topology |

55P60 | Localization and completion in homotopy theory |