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. 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,
∫ M e − t H ω n / n ! = ∑ p e − t H ( p ) t n ∏ α j ( p ) . {\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 α j ( p ) {\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.
History An alternative name for the formula is Borel cohomology, after Armand Borel
Non-abelian localization
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.
References Atiyah, Michael; Raoul, Bott (1984), "The moment map and equivariant cohomology", Topology, 23 (1): 1–28, doi:10.1016/0040-9383(84)90021-1 Liu, Kefeng (2006), "Localization and conjectures from string duality", in Ge, Mo-Lin; Zhang, Weiping (eds.), Differential geometry and physics, Nankai Tracts in Mathematics, vol. 10, World Scientific, pp. 63–105, ISBN 978-981-270-377-4, MR 2322389 Meinrenken, Eckhard (1998), "Symplectic surgery and the Spin c {\displaystyle ^{c}} —Dirac operator", Advances in Mathematics, 134 (2): 240–277, doi:10.1006/aima.1997.1701 Quillen, Daniel (1971), "The spectrum of an equivariant cohomology ring, I", Annals of Mathematics, Second Series, 94 (3): 549–572, doi:10.2307/1970770, JSTOR 1970770; Quillen, Daniel (1971), "The spectrum of an equivariant cohomology ring, II", Annals of Mathematics, Second Series, 94 (3): 573–602, doi:10.2307/1970771, JSTOR 1970771
