nLab Guillemin-Sternberg geometric quantization conjecture

Idea

The Guillemin-Sternberg conjecture states that, given a symplectic manifold equipped with a Hamiltonian action by some group, under suitable conditions the processes of geometric quantization and of symplectic reduction can be interchanged without changing the result (a representation of the group on the space of quantum states). This is usually abbreviated to:

• quantization commutes with reduction.

Since symplectic reduction also means restricting a mechanical system to a constraint surface, one may also read this as saying:

• quantization commutes with imposing constraints.

The original article (Guillemin-Sternberg 82) proved this statement for the case of compact Lie groups acting on compact Kähler manifolds and for geometric quantization via Kähler polarizations.

For the more powerful and more flexible geometric quantization by push-forward, the corresponding statement is widely known as the Guillemin-Sternberg conjecture. Since its formulation, this, too, has been proven in fair generality, see the References below.

Related concepts

• symplectic reduction, presymplectic reduction (gauge reduction), Poisson reduction, BV-BRST formalism

• equivariant cohomology

Reductions of (pre-)symplectic manifolds:

symplectic geometry	physics
presymplectic manifold	covariant phase space
$\downarrow$ gauge reduction	$\downarrow$ quotient by gauge symmetry
symplectic manifold	reduced phase space
$\downarrow$ symplectic reduction	$\downarrow$ quotient by global symmetry
symplectic manifold	reduced phase space

• light-cone gauge quantization

References

The original statement is due to

• Victor Guillemin, Shlomo Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), no. 3, 515–538.

The first proof in geometric quantization by push-forward for compact groups acting on compact manifolds appears in

• Eckhard Meinrenken, Symplectic surgery and the $Spin^c$-Dirac operator, Adv. Math. 134 (1998), 240-277.

A survey is in

• Michèle Vergne, Quantification géométrique et r´éduction symplectique, Séminaire Bourbaki, Vol. 2000/2001. Astérisque No. 282 (2002), Exp. No. 888, viii, 249–278.

Discussion for Kähler polarizations of mildly singular phase spaces is in

• Johannes Huebschmann, Kähler quantization and reduction, J. reine angew. Math. 591 (2006), 75-109 (arXiv:math/0207166)

The generalization to non-compact groups acting on non-compact spaces (but with compact quotients, hence “co-compactly”) by passing to KK-theory and defining the push-forward by the Baum-Connes analytic assembly map was proposed in

• Klaas LandsmanFunctorial quantization and the Guillemin-Sternberg conjecture. In S. Ali, G. Emch, A. Odzijewicz, M. Schlichenmaier, and S. Woronowicz (eds.), Twenty years of Bialowieza: a mathematical anthology, pages 23–45, Singapore, 2005. World Scientific. (arXiv:math-ph/0307059)

and proven for various cases in

• Peter Hochs, Klaas Landsman, The Guillemin-Sternberg conjecture for noncompact groups and spaces (arXiv:math-ph/0512022)

based on

• Peter Hochs, Quantisation commutes with reduction for cocompact Hamiltonian group actions (pdf)

Discussion taking into account the metaplectic correction is in

• Brian Hall, William Kirwin, Unitarity in “quantization commutes with reduction”, Comm. Math. Phys. 275 (2007), no. 3, pages 401 – 442 (arXiv:math/0610005)

The full statement for this general case was then proven in

• Varghese Mathai, Weiping Zhang (appendix by Ulrich Bunke), Geometric quantization for proper actions, Advances in Mathematics

  Volume 225, Issue 3, 20 October 2010, Pages 1224–1247 (arXiv:0806.3138)

Lecture notes on this:

• Weiping Zhang, Analytic aspects of geometric quantization, 2009 (pdf, pdf)

Further generalizations are discussed in

• Peter Hochs, Varghese Mathai, Quantizing tame actions (arXiv:1309.6760)

• Jord Boeijink, Klaas Landsman, Walter van Suijlekom?, Quantization commutes with singular reduction: cotangent bundles of compact Lie groups (arXiv:1508.06763)

Similar discussion is in

• Paul-Emile Paradan, Spin-quantization commutes with reduction (arXiv:0911.1067)

based on

• Paul-Emile Paradan, Localization of the Riemann-Roch character (arXiv:math/9911024)

Discussion specifically for circle group actions is in

• Ana Cannas da Silva, Yael Karshon, Susan Tolman, Quantization of Presymplectic Manifolds and Circle Actions, Transactions of the AMS 352 2 (2000) 525-552 [arXiv:dg-ga/9705008, jstor:118052]