nLab
geometric quantization by push-forward

Idea

Geometric quantization via push-forward or ${\mathrm{Spin}}^{ℂ}$-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.

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

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 $\mathrm{KU}$	third integral SW class $W_3$	spin^c-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

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 ${\mathrm{Spin}}^c$-structures (pdf)

A detailed analysis of psuh-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 ${\mathrm{Spin}}^c$-quantization is in

• Eckhard Meinrenken, Symplectic surgery and the ${\mathrm{Spin}}^c$-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)

Discussion of push-forward not over manifold but over moduli stacks as relevant in Chern-Simons theory is in section 3 of

• Daniel Freed, Michael Hopkins, Constantin Teleman, Consistent Orientation of Moduli Spaces (arXiv:0711.1909)

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