nLab quantization via the A-model



Geometric quantization


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



A method of quantization of mechanical systems represented by symplectic manifolds that produces the space of states of this system as the space of states of open strings in the A-model.

More in detail, the symplectic manifold serves as the target space for the 2-dimensional sigma-model TQFT called the A-model. If the symplectic manifold arises as the complexification of a phase space of some physical system, then one finds that the boundary path integral of the A-model on coisotropic branes computes the path integral and the geometric quantization of this physical system.

The setup is reminiscent of how the deformation quantization of the phase space is computed by the 2-dimensional Poisson sigma-model. See at holographic principle for more on the general pattern.

The goal is to get closer to a systematic theory of quantization.

(Gukov-Witten 08, p.4)


When the phase space MM is the coadjoint orbit for a real form G G_\mathbb{R}, with the complexification being the complex coadjoint orbit YY for G G_\mathbb{C}. The particular cases when G =SL(2,)G_\mathbb{C} = SL(2,\mathbb{C}) are covered in section 3 of (Gukov-Witten 08). These provide applications to representation theory like in the orbit method.


The main original article is

Review includes

Earlier, suggestions that the A-model is related to deformation quantization had been given in

which then have been extended and made more precise in

  • Vasily Pestun, Topological strings in generalized complex space, Adv.Theor.Math.Phys.11:399-450,2007 (arXiv:hep-th/0603145)

and in

in the context of generalized complex geometry.

Tha the space of states of open strings in the A-model can be interpreted as a quantization of the corresponding symplectic manifold had first been noticed in examples in

The canonical coisotropic brane was used to elucidate these matters further in

is the context of the Kapustin-Witten TQFT.

A relation to the path integral is discussed in

Relation to the B-model via mirror symmetry is discussed in

On its relation to geometric quantization:

Last revised on May 3, 2024 at 06:21:12. See the history of this page for a list of all contributions to it.