2004/02/04 by Pascal Lavaud, Lavaud, Pascal
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #math-ph #math.DG #math.MP #math.RT #msc:14M30 #msc:17B70 #msc:58A50 #msc:58C50
paper · pdf · doi:10.48550/arxiv.math/0402068
56 pages
arxiv created 2004/02/04 · arxiv updated 2009/12/01
Let G be a compact Lie group. Let M be a smooth G-manifold and V --> M be an oriented G-equivariant vector bundle. One defines the spaces of equivariant forms with generalized coefficients on V and M. An equivariant Thom form θ on V is a compactly supported closed equivariant form such that its integral along the fibres is the constant function 1 on M. Such a Thom form was constructed by Mathai and Quillen. Its restriction to M gives a representative of the equivariant Euler class of V. In the supergeometric situation we give proper definitions of all the objects involved. But, in this case a Thom form doesn't always exist. In this article, when the action of G on V is sufficiently non-trivial, we construct such a Thom form with generalized coefficients. We use it to construct an equivariant Euler form of V and to generalize Berline-Vergne's localization formula to the supergeometric situation.