In mathematics, more specifically in the context of
geometric quantization, quantization commutes with reduction states that the space of global sections of a line bundle ''L'' satisfying the quantization condition on the
symplectic quotient 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 ...
of a compact
symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
is the space of invariant
sections
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of ''L''.
This was conjectured in 1980s by Guillemin and Sternberg and was proven in 1990s by Meinrenken (the second paper used
symplectic cut In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the symplectic sum, that glues ...
) as well as Tian and Zhang.
For the formulation due to Teleman, see C. Woodward's notes.
See also
*
Geometric invariant theory
Notes
References
*
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Mathematical quantization