nLab geometric quantization by push-forward

Idea

Geometric quantization via push-forward or $Spin^{\mathbb{C}}$-quantization is a variant of geometric quantization in which the step from a prequantum bundle to the space of states is not (explicitly) performed by a choice of polarization and forming the space of polarized sections, but by a choice of spin^c structure and forming the fiber integration in differential K-theory of the prequantum connection. Specifically, this refines the notion of geometric quantization via Kähler polarizations.

Definition

See at geometric quantization the sections

• Quantum state space as index of Dolbeault-Dirac operator

• Quantum state spaces as index of the Spin^c-Dirac operator

Properties

Relation to geometric quantization via Kähler polarization

(…) see (DaSilva-Karshon-Tolman, lemma 2.7, remark 2.9) (…)

Relation to symplectic reduction

(…) see Guillemin-Sternberg geometric quantization conjecture

Related concepts

• metaplectic quantization

• real polarization, Kähler polarization

• almost Kähler quantization

chromatic level	generalized cohomology theory / E-∞ ring	obstruction to orientation in generalized cohomology	generalized orientation/polarization	quantization	incarnation as quantum anomaly in higher gauge theory
1	complex K-theory $KU$	third integral SW class $W_3$	spinᶜ structure	K-theoretic geometric quantization	Freed-Witten anomaly
2	EO(n)	Stiefel-Whitney class $w_4$
2	integral Morava K-theory $\tilde K(2)$	seventh integral SW class $W_7$			Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation

References

A survey is in

• Reyer Sjamaar, Symplectic reduction and Riemann-Roch formulas for multiplicities, Bull. Amer. Math. Soc. 33 (1996), 327-338 (AMS)

The idea originates around

• David Metzler, A K-Theoretic Note on Geometric Quantization (arXiv:math/9809031)

based on

• Hans Duistermaat, Victor Guillemin, Eckhard Meinrenken, Siye Wu, Symplectic reduction and Riemann-Roch for circle actions, Mathematical Research Letters 2, 259–266 (1995) (pdf)

and is highlighted in the general context of geometric quantization in

• Viktor Ginzburg, Victor Guillemin, Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, AMS (2002)

  (Section 6.8 “Geometric quantization as a push-forward”)

and the last section of

• Stable complex and $Spin^c$-structures (pdf)

A detailed analysis of push-forward quantization of general presymplectic manifolds is in

• Ana Canas da Silva, Yael Karshon, Susan Tolman, Quantization of Presymplectic Manifolds and Circle Actions, Trans. Amer. Math. Soc. 352 (2000), 525-552 (arXiv:dg-ga/9705008)

A first proof of the Guillemin-Sternberg geometric quantization conjecture in terms of $Spin^c$-quantization is in

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

A suggestion that geometric push-forward quantization is best understood to proceed to take values in KK-theory is in

• Klaas Landsman, Functorial quantization and the Guillemin-Sternberg conjecture, Proc. Bialowieza 2002 (arXiv:math-ph/0307059)

• Rogier Bos, Groupoids in geometric quantization PhD Thesis (2007) (pdf)

A refined realization of the Guillemin-Sternberg geometric quantization conjecture was conjectured in

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

based on the thesis

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

and was proven in

• Varghese Mathai, Weiping Zhang, Geometric quantization for proper actions, Advances in Mathematics 225:1224-1247 (2010) (arXiv:0806.3138)

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 of push-forward not over manifolds but over moduli stacks as relevant in Chern-Simons theory is in

• Daniel Freed, Michael Hopkins, Constantin Teleman, Consistent Orientation of Moduli Spaces (arXiv:0711.1909), chapter XIX, pages 395-419 in: Oscar Garcia-Prada, Jean Pierre Bourguignon, Simon Salamon (eds.) The Many Facets of Geometry: A Tribute to Nigel Hitchin, Oxford University Press 2010 (doi:10.1093/acprof:oso/9780199534920.001.0001)

• Shay Fuchs, $Spin^{\mathbb{C}}$-quantization, prequantization and cutting, Ph.D. thesis, University of Toronto (2008) (pdf)

Discussion of this push-forward quantization as higher geometric quantization of symplectic groupoids is in

• Joost Nuiten, Cohomological quantization of local prequantum boundary field theory, MSc thesis, Utrecht, August 2013 (thesis pdf, archived version, talk slides)