geometric quantization by push-forward



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Geometric quantization via push-forward or Spin 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 at geometric quantization the sections


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

chromatic levelgeneralized cohomology theory / E-∞ ringobstruction to orientation in generalized cohomologygeneralized orientation/polarizationquantizationincarnation as quantum anomaly in higher gauge theory
1complex K-theory KUKUthird integral SW class W 3W_3spin^c-structureK-theoretic geometric quantizationFreed-Witten anomaly
2EO(n)Stiefel-Whitney class w 4w_4
2integral Morava K-theory K˜(2)\tilde K(2)seventh integral SW class W 7W_7Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation


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

based on

and is highlighted in the general context of geometric quantization in

and the last section of

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

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

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

  • Eckhard Meinrenken, Symplectic surgery and the Spin cSpin^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

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

based on the thesis

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

and was proven in

Similar discussion is in

based on

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

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

Revised on August 10, 2017 13:51:55 by Urs Schreiber (