Localization formula for equivariant cohomology

In differential geometry, the localization formula states: for an equivariantly closed equivariant differential form α on an orbifold M with a torus action and for a sufficient small ξ in the Lie algebra of the torus T,

where the sum runs over all connected components F of the set of fixed points M T , d M is the orbifold multiplicity of M (which is one if M is a manifold) and e T ( F ) is the equivariant Euler form of the normal bundle of 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. 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 α j ( p ) 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 Quillen's papers.