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

Contents

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

previous chapters: modules


Contents

Introduction

In the section prequantized Lagrangian correspondences we had seen how classical mechanics is captured by smooth groupoids slices over BU(1)\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 B 2U(1)\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,π)SG(X,\pi) of a Poisson manifold (X,π)(X,\pi) as equipped with its prequantum line 2-bundle. SH(X,π)B 2U(1)SH(X,\pi) \to \mathbf{B}^2 U(1). We linearize this to an (infinity,1)-module bundle by composing with B 2U(1)BGL 1(KU)\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.

X * SG(X,π) prequantum B 2U(1) quantization BGL 1(KU) KU(*) KU(SG(X,π)) quantum \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 0H_0 for the /2\mathbb{Z}/2\mathbb{Z}-graded separable Hilbert space.

Proposition

For XX a smooth manifold and GG a compact Lie group with action on XX, then the Hilbert bundle

EX×H 0L 2(G)X E \coloneqq X \times H_0 \otimes L^2(G) \to X

equipped with the GG-action on the fibers given on L 2(G)L^2(G) by pullback along the right GG-action on itself is a universal equivariant Hilbert bundle, meaning that the space of equivariant sections of the associated Fredholm bundle is the GG-equivariant K-theory of XX

K (X//G)Γ(X,Fred (E)). 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 GG-action

(Nuiten, section 5.2.1)

Universal orbit quantization

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

References

Last revised on April 16, 2015 at 09:51:52. See the history of this page for a list of all contributions to it.