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geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
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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.
When the phase space $M$ is the coadjoint orbit for a real form $G_\mathbb{R}$, with the complexification being the complex coadjoint orbit $Y$ for $G_\mathbb{C}$. The particular cases when $G_\mathbb{C} = SL(2,\mathbb{C})$ 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
Paul Bressler, Yan Soibelman, Mirror symmetry and deformation quantization (arXiv:hep-th/0202128)
Anton Kapustin, A-branes and Noncommutative Geometry (arXiv:hep-th/0502212)
which then have been extended and made more precise in
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:
Davide Gaiotto, Edward Witten. Probing Quantization Via Branes (2021). (arXiv:2107.12251)
Joshua Lackman, Geometric Quantization Without Polarizations [arXiv:2405.01513]
Last revised on May 3, 2024 at 06:21:12. See the history of this page for a list of all contributions to it.