This entry is a chapter of geometry of physics. See there for context.
previous chapters: modules
In the section prequantized Lagrangian correspondences we had seen how classical mechanics is captured by smooth groupoids slices over $\mathbf{B}U(1)$.
We consider this now as the boundary field theory of the 2d Poisson-Chern-Simons theory in smooth infinity-groupoids over $\mathbf{B}^2 U(1)$ and discuss geometric quantization from the point of view of the holographic/motivic quantization of this system.
That is we consider the symplectic groupoid $SG(X,\pi)$ of a Poisson manifold $(X,\pi)$ as equipped with its prequantum line 2-bundle. $SH(X,\pi) \to \mathbf{B}^2 U(1)$. We linearize this to an (infinity,1)-module bundle by composing with $\mathbf{B}^2 U(1)\to B GL_1(KU)$. Then we perform the boundary path integral quantization by fiber integration in K-theory and show how this reproduces, on the boundary, traditional geometric quantization such as in particular the orbit method.
The abstract picture behind this cohomological motivic quantization is discussed at dependent linear type theory. Here we focus on explicit details for the case of KU-quantization.
(Rosenberg 04, Atiyah-Segal 04), review is in (Nuiten section 3.2.1)
(FHT I), reviewed in (Nuiten section 3.3.2)
Write $H_0$ for the $\mathbb{Z}/2\mathbb{Z}$-graded separable Hilbert space.
For $X$ a smooth manifold and $G$ a compact Lie group with action on $X$, then the Hilbert bundle
equipped with the $G$-action on the fibers given on $L^2(G)$ by pullback along the right $G$-action on itself is a universal equivariant Hilbert bundle, meaning that the space of equivariant sections of the associated Fredholm bundle is the $G$-equivariant K-theory of $X$
along maps of manifolds (CareyWang08)
along representable morphisms of local quotient stacks (FHT I)
review is in Nuiten section 4.2
(FHT II, section 1) (Nuiten, section 5.2.2)
Stephan Bongers, Geometric quantization of symplectic and Poisson manifolds, 2014
Joost Nuiten, Cohomological quantization of local prequantum boundary field theory, 2013
Alan Carey, Bai-Ling Wang, Thom isomorphism and push-forward map in twisted K-theory, J. K-Theory 1(2), 357-393 (2008) (arXiv:0507414)
Daniel Freed, Mike Hopkins, Constantin Teleman, Loop Groups and Twisted K-Theory I (2011) (arXiv:0711.1906), II (2013) (arXiv:0511232)
Last revised on April 16, 2015 at 09:51:52. See the history of this page for a list of all contributions to it.