Equivariant Cohomology and Localization Formula in Supergeometry

LetG be a compact Lie group. LetM be a smoothG-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 [MQ86]. 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 ([BV83a]) to the supergeometric situation. The aim of this article is to generalize Berline-Vergne’s localization formula ([BV83a]) to the supergeometric situation. Let G = (G0, g) be a Lie supergroup (with underlying Lie group G0 and Lie superalgebra g) acting on the right on a globally oriented supermanifold M = (M0,OM) (see section 2.4 for precise definitions). Let α be a closed equivariant integrable form (the superanalog of closed equivariant forms with compact support). Let X be an element of g0 such that exp(RX) is a compact subgroup of G0. Let j : M(X) →֒ M be the subsupermanifold of zeroes of X. Under some conditions, we construct an equivariant Euler form for TNM(X) (the normal bundle of M(X) in M). We denote it by Eg. Then we obtain, for Z in a neighborhood of X in g(X) (the centralizer of X in g) the following equality: