Quantization commutes with reduction

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^[1] on the symplectic quotient of a compact symplectic manifold is the space of invariant sections^[vague] of L.

This was conjectured in 1980s by Guillemin and Sternberg and was proven in 1990s by Meinrenken^[2][3] (the second paper used symplectic cut) as well as Tian and Zhang.^[4] For the formulation due to Teleman, see C. Woodward's notes.

See also

• Geometric invariant theory