Equivariant Cohomology and Localization Formula in Supergeometry

TL;DR

This paper extends the concepts of equivariant cohomology and localization formulas to supergeometry, providing new constructions of Thom forms and Euler classes in this generalized setting, especially under non-trivial group actions.

Contribution

It introduces a proper supergeometric framework for equivariant forms and constructs Thom forms and localization formulas where they previously did not exist.

Findings

Constructed equivariant Thom forms with generalized coefficients in supergeometry.

Generalized Berline-Vergne localization formula to supergeometric context.

Provided conditions for existence of Thom forms under non-trivial group actions.

Abstract

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 $\theta$ 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.