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In differential geometry, the localization formula states: 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,
where the sum runs over all connected components F of the set of fixed points M T {\displaystyle M^{T}} , d M {\displaystyle d_{M}} is the orbifold multiplicity of M and e T {\displaystyle e_{T}} 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 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 on a compact symplectic manifold M of dimension 2n,