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A Simple Description of the Hyperkähler Structure of the Cotangent Bundle of Projective Space via Quantization

Joshua Lackman

TL;DR

The paper develops a coordinate-free, quantization-based description of the hyperkähler geometry on the cotangent bundle of projective space by identifying $T^*\mathbb{P}\mathcal{H}$ with all rank-1 projections inside $\mathcal{B}(\mathcal{H})$ and endowing it with a metric whose Kahler potential is the Hilbert–Schmidt norm. It constructs a rich family of complex structures, including commuting and anticommuting types, and a holomorphic symplectic form that together yield generalized Kahler and hyperkähler structures, with explicit formulas such as $\Omega(A,B)=i\mathrm{Tr}(q[A,B])$ and $JA=i[A,q]$. It also develops a Poisson-geometric and path-integral quantization framework, including a fiberwise Toeplitz quantization via a canonical idempotent section, and describes a natural compactification and extensions to Grassmannians. In the $\mathcal{H}=\mathbb{C}^2$ case, the hyperkähler metric recovers Eguchi–Hanson, and the construction generalizes through Plücker embeddings to broader Grassmannian contexts, offering a coordinate-free approach to the quantized hyperkähler geometry of these spaces.

Abstract

Quantization identifies the cotangent bundle of projective space with the (non-Hermitian) rank-$1$ projections of a Hilbert space. We use this identification to study the natural geometric structures of these cotangent bundles and those of Grassmanians. In particular, we show that the quantization map is an isometric and complex embedding $T^*\mathbb{P}\mathcal{H}\hookrightarrow\mathcal{B}(\mathcal{H})\backslash\{0\}.$ Here, the metric on the domain is the hyperkähler metric and the metric on the codomain is the one whose Kähler potential is the Hilbert-Schmidt norm. The Kähler potential pulled back to $T^*\mathbb{P}\mathcal{H}$ equals the trace-class norm. Using this, we give a complete, simple and explicit description of the hyperkähler structure. Our constructions are functorial, coordinate-free and reduction-free.

A Simple Description of the Hyperkähler Structure of the Cotangent Bundle of Projective Space via Quantization

TL;DR

The paper develops a coordinate-free, quantization-based description of the hyperkähler geometry on the cotangent bundle of projective space by identifying with all rank-1 projections inside and endowing it with a metric whose Kahler potential is the Hilbert–Schmidt norm. It constructs a rich family of complex structures, including commuting and anticommuting types, and a holomorphic symplectic form that together yield generalized Kahler and hyperkähler structures, with explicit formulas such as and . It also develops a Poisson-geometric and path-integral quantization framework, including a fiberwise Toeplitz quantization via a canonical idempotent section, and describes a natural compactification and extensions to Grassmannians. In the case, the hyperkähler metric recovers Eguchi–Hanson, and the construction generalizes through Plücker embeddings to broader Grassmannian contexts, offering a coordinate-free approach to the quantized hyperkähler geometry of these spaces.

Abstract

Quantization identifies the cotangent bundle of projective space with the (non-Hermitian) rank- projections of a Hilbert space. We use this identification to study the natural geometric structures of these cotangent bundles and those of Grassmanians. In particular, we show that the quantization map is an isometric and complex embedding Here, the metric on the domain is the hyperkähler metric and the metric on the codomain is the one whose Kähler potential is the Hilbert-Schmidt norm. The Kähler potential pulled back to equals the trace-class norm. Using this, we give a complete, simple and explicit description of the hyperkähler structure. Our constructions are functorial, coordinate-free and reduction-free.

Paper Structure

This paper contains 14 sections, 44 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Differential Geometry of $\textup{T}^*\mathbf{G}\mathcal{H}$
  3. $\textup{T}^*\mathbf{G}\mathcal{H}$ as an Affine Subvariety of $\mathcal{B}(\mathcal{H})$
  4. Important Identities
  5. Differential Geometry of $\textup{T}^*\mathbf{G}\mathcal{H}$
  6. Two Commuting Complex Structures, Symplectic Form and Exterior Derivative
  7. Vector Bundle Structure
  8. Projection Map
  9. Vector Space Structure of Fibers and Compactification
  10. The Connection and Third Commuting Complex Structure
  11. Compactification of $\textup{T}^*\mathbf{G}\mathcal{H}$ and Another Description of $I,J,\hat{J}$
  12. Poisson Geometry
  13. The Idempotent Section (Path Integral)
  14. The Hyperkähler Structure of $\textup{T}^*\mathbb{P}\mathcal{H}$

Key Result

Proposition 1

Assume the embedding $\textup{T}^*\mathbb{P}\mathcal{H}\xhookrightarrow{}\mathcal{B}(\mathcal{H})\backslash\{0\}.$ We have a hyperkähler structure given as follows: the Hilbert–Schmidt norm $\textup{T}^*\mathbb{P}\mathcal{H}\to\mathbb{R},\,q\mapsto\|q\|$ is a Kähler potential for the Riemannian metr where $A,B\in \textup{T}_q\textup{T}^*\mathbb{P}\mathcal{H}\subset \mathcal{B}(\mathcal{H})$ and th

Theorems & Definitions (102)

  • Proposition 1
  • Remark 2
  • Lemma 1.0.1
  • proof
  • Proposition 1.0.2
  • proof
  • Proposition 1.0.3
  • proof
  • Proposition 1.0.4
  • proof
  • ...and 92 more