# nLab geometric quantization by push-forward

## Surveys, textbooks and lecture notes

integration

### Variants

#### Differential cohomology

differential cohomology

# Contents

## 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.

## Properties

### Relation to geometric quantization via Kähler polarization

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

### Relation to symplectic reduction

chromatic levelgeneralized cohomology theory / E-∞ ringobstruction to orientation in generalized cohomologygeneralized orientation/polarizationquantizationincarnation as quantum anomaly in higher gauge theory
1complex K-theory $KU$third integral SW class $W_3$spin^c-structureK-theoretic geometric quantizationFreed-Witten anomaly
2EO(n)Stiefel-Whitney class $w_4$
2integral 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

based on

and is highlighted in the general context of geometric quantization in

and the last section of

• Stable complex and $Spin^c$-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 $Spin^c$-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

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 manifold but over moduli stacks as relevant in Chern-Simons theory is in section 3 of

Revised on September 21, 2014 14:10:41 by Urs Schreiber (185.26.182.32)