Equivariant Localization

Equivariant Localization

The purpose of this note is to construct and characterize localizations of G-spaces, where G is a compact Lie group. It was written at the request of Reinhard Schultz, who needed its results in work of his own. When G is finite, the possibility of such a construction is folklore familiar to the experts. That the general case presents little more difficulty comes as something of a surprise. Of c...

Equivariant Localization and Mirror Symmetry

Equivariant Localization of Path Integrals

We review equivariant localization techniques for the evaluation of Feynman path integrals. We develop systematic geometric methods for studying the semi-classical properties of phase space path integrals for dynamical systems, emphasizing the relations with integrable and topological quantum field theories. Beginning with a detailed review of the relevant mathematical background – equivariant ...

On the localization formula in equivariant cohomology

We give a generalization of the Atiyah–Bott–Berline–Vergne localization theorem for the equivariant cohomology of a torus action. We replace the manifold having a torus action by an equivariant map of manifolds having a compact connected Lie group action. This provides a systematic method for calculating the Gysin homomorphism in ordinary cohomology of an equivariant map. As an example, we reco...

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 an...