In differential geometry, the localization formula states: for an equivariantly closed equivariant differential form ${\displaystyle \alpha }$ on an orbifold M with a torus action and for a sufficient small ${\displaystyle \xi }$ in the Lie algebra of the torus T,

${\displaystyle {1 \over d_{M}}\int _{M}\alpha (\xi )=\sum _{F}{1 \over d_{F}}\int _{F}{\alpha (\xi ) \over e_{T}(F)(\xi )}}$

where the sum runs over all connected components F of the set of fixed points ${\displaystyle M^{T}}$, ${\displaystyle d_{M}}$ is the orbifold multiplicity of M (which is one if M is a manifold) and ${\displaystyle e_{T}(F)}$ 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,

${\displaystyle \int _{M}e^{-tH}\omega ^{n}/n!=\sum _{p}{e^{-tH(p)} \over t^{n}\prod \alpha _{j}(p)}.}$

where H is Hamiltonian for the circle action, the sum is over points fixed by the circle action and ${\displaystyle \alpha _{j}(p)}$ 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 Daniel Quillen's papers.