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In differential geometry, the localization … In differential geometry, the localization formula states: for an equivariantly closed equivariant differential form on an orbifold M with a torus action and for a sufficient small in the Lie algebra of the torus T, where the sum runs over all connected components F of the set of fixed points , is the orbifold multiplicity of M (which is one if M is a manifold) and 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, where H is Hamiltonian for the circle action, the sum is over points fixed by the circle action and 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 , 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.s is discussed in Daniel Quillen's papers.
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| rdfs:comment |
In differential geometry, the localization … In differential geometry, the localization formula states: for an equivariantly closed equivariant differential form on an orbifold M with a torus action and for a sufficient small in the Lie algebra of the torus T, where the sum runs over all connected components F of the set of fixed points , is the orbifold multiplicity of M (which is one if M is a manifold) and is the equivariant Euler form of the normal bundle of F. where H is Hamiltonian for the circle action, the sum is over points fixed by the circle action and are eigenvalues on the tangent space at p (cf. Lie group action.)tangent space at p (cf. Lie group action.)
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Localization formula for equivariant cohomology
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