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
- ↑ This means that the curvature of the connection on the line bundle is the symplectic form.
- ↑ Meinrenken 1996
- ↑ Meinrenken 1998
- ↑ Tian−Zhang 1998
References
- V. Guillemin, S. Sternberg, Geometric quantization and multiplicities of group representations, Inventiones mathematicae 1982, Volume 67, Issue 3, pp 515–538
- E. Meinrenken, On Riemann–Roch formulas for multiplicities, Journal of the A.M.S. 9 (1996), 373–389.
- E. Meinrenken, Symplectic surgery and the Spinc–Dirac operator, Adv. Math. 134 (1998), no. 2, 240–277.
- Y. Tian and W. Zhang, An analytic proof of the geometric quantization conjecture of Guillemin–Sternberg, Invent. Math. 132, 229–259 (1998)
- Woodward, Christopher T. (2011). "Moment maps and geometric invariant theory". arXiv:0912.1132.