Supergeometry in Equivariant Cohomology

TL;DR

This paper explores $S^1$ equivariant cohomology through supergeometry, introducing superstructures and deriving localization formulas using superanalogues of classical theorems.

Contribution

It introduces a supergeometric framework for equivariant cohomology, connecting superstructures with classical invariants and localization techniques.

Findings

Supergeometric structures encode equivariant Euler classes.

Localization formulas are derived via superanalogues of Stokes theorem.

Equivariant cohomology is analyzed using supergeometry methods.

Abstract

We analyze $S^1$ equivariant cohomology from the supergeometrical point of view. For this purpose we equip the external algebra of given manifold with equivariant even super(pre)symplectic structure, and show, that its Poincare-Cartan invariant defines equivariant Euler classes of surfaces. This allows to derive localization formulae by use of superanalog of Stockes theorem.