nLab geometry of physics -- geometric quantization with KU-coefficients

This entry is a chapter of geometry of physics. See there for context.

previous chapters: modules

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Introduction

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.

$$\array{ && X \\ & \swarrow && \searrow \\ \ast && && SG(X,\pi) && && pre-quantum \\ & \searrow & \swArrow & \swarrow \\ && \mathbf{B}^2 U(1) \\ && \downarrow && && && \downarrow^{\mathrlap{quantization}} \\ && B GL_1(KU) \\ \\ KU(\ast) &&\longrightarrow && KU(SG(X,\pi)) && && quantum }$$

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.

KU-theory

Twisted KU-cohomology of manifolds

(Rosenberg 04, Atiyah-Segal 04), review is in (Nuiten section 3.2.1)

Twisted KU-cohomology of smooth local quotient stacks

(FHT I), reviewed in (Nuiten section 3.3.2)

Write $H_0$ for the $\mathbb{Z}/2\mathbb{Z}$-graded separable Hilbert space.

Proposition

For $X$ a smooth manifold and $G$ a compact Lie group with action on $X$, then the Hilbert bundle

$$E \coloneqq X \times H_0 \otimes L^2(G) \to X$$

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 $X$

$$K^\bullet(X//G) \simeq \Gamma(X, Fred^\bullet(E)) \,.$$

Twisted integration with KU-coefficients

fiber integration in K-theory

along maps of manifolds (CareyWang08)

along representable morphisms of local quotient stacks (FHT I)

review is in Nuiten section 4.2

Geometric quantization

Bongers, section 2

Symplectic manifold with Hamiltonian $G$-action

(Nuiten, section 5.2.1)

Universal orbit quantization

(FHT II, section 1) (Nuiten, section 5.2.2)

References

• 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)