Localization formula for equivariant cohomology

In differential geometry, the localization formula states that for an equivariantly closed equivariant differential form α {\displaystyle \alpha } on an orbifold M with a torus action and for a sufficient small ξ {\displaystyle \xi } in the Lie algebra of the torus T, we have 1 d M ∫ M α ( ξ ) = ∑ F 1 d F ∫ F α ( ξ ) e T ( F ) ( ξ ) {\displaystyle {1 \over d_{M}}\int _{M}\alpha (\xi )=\sum _{F}{1 \over d_{F}}\int _{F}{\alpha (\xi ) \over e_{T}(F)(\xi )}} where the sum runs over all connected components F of the set M T {\displaystyle M^{T}} of fixed points, d M {\displaystyle d_{M}} is the orbifold multiplicity of M {\displaystyle M} (which equals 1 {\displaystyle 1} if M {\displaystyle M} is a manifold), and e T ( F ) {\displaystyle e_{T}(F)} is the equivariant Euler form of the normal bundle of F {\displaystyle F} . The formula allows one to compute the equivariant cohomology ring of the orbifold M (a particular kind of differentiable stack) from the equivariant cohomology of its fixed point components, up to multiplicities and Euler forms.

Source: Wikipedia — Localization formula for equivariant cohomology (CC BY-SA 4.0)