nLab
geometric quantization by push-forward

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Integration theory

Differential cohomology

Contents

Idea

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.

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

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

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

Revised on July 13, 2013 02:04:31 by Urs Schreiber (89.204.137.126)