In mathematics, the Duistermaat–Heckman formula, due to , states that the
pushforward of the canonical (
Liouville
Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
Life and work
He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...
) measure on a
symplectic manifold under the
moment map In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the ac ...
is a piecewise polynomial measure. Equivalently, the
Fourier transform of the canonical measure is given ''exactly'' by the
stationary phase approximation In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to the limit as k \to \infty .
This method originates from the 19th century, and is due to George Gabriel Stokes and Lord Kelvin.
It is closel ...
.
and, independently, showed how to deduce the Duistermaat–Heckman formula from a
localization theorem
In mathematics, particularly in integral calculus, the localization theorem allows, under certain conditions, to infer the nullity of a function given only information about its continuity and the value of its integral.
Let be a real-valued fun ...
for
equivariant cohomology In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordi ...
.
References
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External links
*http://terrytao.wordpress.com/2013/02/08/the-harish-chandra-itzykson-zuber-integral-formula/
{{DEFAULTSORT:Duistermaat-Heckman formula
Symplectic geometry