The twistor space of ${\mathbb{R}}^{4n}$ and Berezin-Toeplitz operators

Abstract

A hyperkähler manifold $M$ has a family of induced complex structures indexed by a two-dimensional sphere $S^2\cong {\mathbb{CP}}^1$. The twistor space of $M$ is a complex manifold $Tw(M)$ together with a natural holomorphic projection $Tw(M)\to {\mathbb{CP}}^1$, whose fiber over each point of ${\mathbb{CP}}^1$ is a copy of $M$ with the corresponding induced complex structure. We remove one point from this sphere (corresponding to one fiber in the twistor space),and for the case of $M={\mathbb{R}}^{4n}$, $n\in \mathbb{N}$, equipped with the standard hyperkähler structure, we construct one quantization that replaces the family of Berezin-Toeplitz quantizations parametrized by $S^2-\{pt\}$. We provide semiclassical asymptotics for this quantization.

arXiv comment23 pages; minor changes