A Simple Description of the Hyperk\"{a}hler Structure of the Cotangent Bundle of Projective Space via Quantization

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 hyperk\"{a}hler metric and the metric on the codomain is the one whose K\"{a}hler potential is the Hilbert-Schmidt norm. The K\"{a}hler 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 hyperk\"{a}hler structure. Our constructions are functorial, coordinate-free and reduction-free.