Localization formula for equivariant cohomology
Template:Short description In differential geometry, the localization formula states that for an equivariantly closed equivariant differential form on an orbifold M with a torus action and for a sufficient small in the Lie algebra of the torus T, we have
where the sum runs over all connected components F of the set of fixed points, is the orbifold multiplicity of (which equals if is a manifold), and is the equivariant Euler form of the normal bundle of .
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. No analog of such results holds in the non-equivariant cohomology.
One important consequence of the formula is the Duistermaat–Heckman theorem, which states: supposing there is a Hamiltonian circle action (for simplicity) on a compact symplectic manifold M of dimension 2n,
where H is Hamiltonian for the circle action, the sum is over points fixed by the circle action and are eigenvalues on the tangent space at p (cf. Lie group action.)
The localization formula can also computes the Fourier transform of (Kostant's symplectic form on) coadjoint orbit, yielding the Harish-Chandra's integration formula, which in turns gives Kirillov's character formula.
The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers.
Non-abelian localization
Template:Expand section The localization theorem states that the equivariant cohomology can be recovered, up to torsion elements, from the equivariant cohomology of the fixed point subset. This does not extend, in verbatim, to the non-abelian action. But, there is still a version of the localization theorem for non-abelian actions.