Duistermaat–Heckman formula
E895659
The Duistermaat–Heckman formula is a result in symplectic geometry that describes how the pushforward of the Liouville measure under a moment map behaves, showing it is piecewise polynomial and linking geometry with equivariant localization techniques.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Duistermaat–Heckman formula canonical | 1 |
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in symplectic geometry ⓘ |
| appliesTo |
Hamiltonian actions of compact Lie groups on symplectic manifolds
ⓘ
Hamiltonian torus actions ⓘ |
| assumes | compactness conditions on the symplectic manifold or properness of the moment map ⓘ |
| characterizes | pushforward of symplectic volume under the moment map ⓘ |
| connectedTo |
Atiyah–Bott localization theorem
NERFINISHED
ⓘ
Berline–Vergne localization formula NERFINISHED ⓘ Kirwan convexity theorem NERFINISHED ⓘ moment polytope ⓘ |
| describes | behavior of the pushforward of the Liouville measure under a moment map ⓘ |
| field |
Hamiltonian group actions
ⓘ
equivariant cohomology ⓘ symplectic geometry ⓘ |
| hasApplicationIn |
geometric quantization
ⓘ
integrable systems ⓘ mathematical physics ⓘ representation theory of compact Lie groups ⓘ |
| implies |
density of the pushforward measure is polynomial on each chamber of regular values of the moment map
ⓘ
jumps in the density occur when crossing walls of singular values of the moment map ⓘ |
| inspired | developments in equivariant localization techniques ⓘ |
| involvesConcept |
Fourier transform of symplectic volume
ⓘ
Liouville measure NERFINISHED ⓘ coadjoint orbits ⓘ equivariant localization ⓘ fixed points of group actions ⓘ moment map ⓘ piecewise polynomial density ⓘ symplectic volume ⓘ |
| language | mathematical English ⓘ |
| namedAfter |
Gert Heckman
NERFINISHED
ⓘ
Johannes J. Duistermaat NERFINISHED ⓘ |
| originalAuthors |
Gert Heckman
NERFINISHED
ⓘ
Johannes J. Duistermaat NERFINISHED ⓘ |
| originalPublication | Acta Mathematica NERFINISHED ⓘ |
| relates |
geometry of moment map to combinatorics of polytopes
ⓘ
symplectic geometry to equivariant cohomology ⓘ symplectic geometry to representation theory ⓘ |
| states | the pushforward of the Liouville measure by the moment map has a piecewise polynomial density ⓘ |
| typeOf | localization formula ⓘ |
| usedFor |
computing distributions of values of the moment map
ⓘ
computing symplectic volumes of reduced spaces ⓘ studying Hamiltonian torus actions on compact symplectic manifolds ⓘ studying symplectic reduction ⓘ |
| yearProved | 1982 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.