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

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, 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 Spin^c-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