Localization formula for equivariant cohomology

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

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

) 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

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

.

The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers.

Non-abelian localization

This section needs expansion. You can help by adding to it. (November 2014)

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

• doi:10.1016/0040-9383(84)90021-1
• MR 2322389
• Meinrenken, Eckhard (1998), "Symplectic surgery and the Spin${\displaystyle ^{c}}$—Dirac operator",
  doi:10.1006/aima.1997.1701
• JSTOR 1970771

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