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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.
(…) see (DaSilva-Karshon-Tolman, lemma 2.7, remark 2.9) (…)
(…) see Guillemin-Sternberg geometric quantization conjecture
A survey is in
The idea originates around
based on
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
A detailed analysis of psuh-forward quantization of general presymplectic manifolds is in
A first proof of the Guillemin-Sternberg geometric quantization conjecture in terms of $Spin^c$-quantization is in
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
based on the thesis
and was proven in
Similar discussion is in
based on
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, $Spin^{\mathbb{C}}$-quantization, prequantization and cutting Ph.D. thesis, University of Toronto (2008) (pdf)