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 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 (the second paper used symplectic cut) as well as Tian and Zhang. For the formulation due to Teleman, see C. Woodward's notes.
See also
Notes
- 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, S2CID 121632102
- 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, S2CID 119943992.
- Woodward, Christopher T. (2010), "Moment maps and geometric invariant theory", Les Cours du CIRM, 1 (1): 55–98, arXiv:0912.1132, Bibcode:2009arXiv0912.1132W, doi:10.5802/ccirm.4
References
This means that the curvature of the connection on the line bundle is the symplectic form. /wiki/Curvature_form ↩
Meinrenken 1996 - 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 https://doi.org/10.1090%2FS0894-0347-96-00197-X ↩
Meinrenken 1998 - 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 https://arxiv.org/abs/dg-ga/9504002 ↩
Tian & Zhang 1998 - 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, S2CID 119943992 https://ui.adsabs.harvard.edu/abs/1998InMat.132..229T ↩