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 ${{\,\mathrm{Tw}\,}}(M)$ together with a natural holomorphic projection ${{\,\mathrm{Tw}\,}}(M) \rightarrow \mathbb {CP}^1$, 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.