In differential geometry, the localization formula states: for an equivariantly closed equivariant differential form

\alpha

on an orbifold ''M'' with a

torus action In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus ''T'' is called a ''T''-variety. In differential geometry, one considers an action of ...

and for a sufficient small

\xi

in the Lie algebra of the torus ''T'', :

\int_M \alpha(\xi) = \sum_F  \int_F

where the sum runs over all connected components ''F'' of the set of fixed points

M^T

d_M

is the

orbifold multiplicity In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...

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 A differentiable stack is the analogue in differential geometry of an algebraic stack in algebraic geometry. It can be described either as a stack over differentiable manifolds which admits an atlas, or as a Lie groupoid up to Morita equivalence. D ...

) 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 2''n'', :

\int_M e^ \omega^n/n! = \sum_p .

where ''H'' is Hamiltonian for the circle action, the sum is over points fixed by the circle action and

\alpha_j(p)

are eigenvalues on the tangent space at ''p'' (cf. Lie group action.) The localization formula can also computes the

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

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.

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

* * * *; Differential geometry {{differential-geometry-stub