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 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

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
  • Guillemin, V.; Sternberg, S. (1982), "Geometric quantization and multiplicities of group representations", Inventiones Mathematicae, 67 (3): 515–538, Bibcode:1982InMat..67..515G, doi:10.1007/BF01398934, MR 0664118
  • Meinrenken, Eckhard (1996), "On Riemann-Roch formulas for multiplicities", Journal of the American Mathematical Society, 9 (2): 373–389, doi:10.1090/S0894-0347-96-00197-X, MR 1325798.
  • Meinrenken, Eckhard (1998), "Symplectic surgery and the Spinc-Dirac operator", Advances in Mathematics, 134 (2): 240–277, arXiv:dg-ga/9504002, doi:10.1006/aima.1997.1701, MR 1617809.
  • Tian, Youliang; Zhang, Weiping (1998), "An analytic proof of the geometric quantization conjecture of Guillemin–Sternberg", Inventiones Mathematicae, 132 (2): 229–259, Bibcode:1998InMat.132..229T, doi:10.1007/s002220050223, MR 1621428.
  • Woodward, Christopher T. (2011), Moment maps and geometric invariant theory, arXiv:0912.1132, Bibcode:2009arXiv0912.1132W


This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.