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 satisfying the quantization condition[1] on the symplectic quotient of a compact symplectic manifold is the space of invariant sections of the line bundle.

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

Notes

  1. This means that the curvature of the connection on the line bundle is the symplectic form.
  2. Meinrenken 1996
  3. Meinrenken 1998
  4. Tian & Zhang 1998

References


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