A Simple Description of the Hyperkähler Structure of the Cotangent Bundle of Projective Space via Quantization
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912273816092672 |
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| author | Lackman, Joshua |
| author_facet | Lackman, Joshua |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10570 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Simple Description of the Hyperkähler Structure of the Cotangent Bundle of Projective Space via Quantization Lackman, Joshua Symplectic Geometry High Energy Physics - Theory Mathematical Physics Algebraic Geometry Quantum Algebra 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. |
| title | A Simple Description of the Hyperkähler Structure of the Cotangent Bundle of Projective Space via Quantization |
| topic | Symplectic Geometry High Energy Physics - Theory Mathematical Physics Algebraic Geometry Quantum Algebra |
| url | https://arxiv.org/abs/2503.10570 |