twistor_space
Let $X$ be a manifold with holonomy $G$, and let $(\rho,V)$ be a complex irreducible representation of $G$. Twistor space $Z_{X,\rho}$ is defined to be the fibrewise projectivization of the associated bundle $P \times_G V$, where $P$ is the holonomy bundle of $X$ (a principal $G$-bundle). Thus, elements of $Z_{X,\rho}$ are equivalence classes $[p; v]$ of $p \in P, 0 \neq v \in V$ where $[pg; g^{-1} \lambda v]$ is equivalent to $[p, v]$ for all non-zero $\lambda \in \mathbb{C}$ and $g \in G$. The dimension of $Z_{X,\rho}$ is $\dim X + \dim_{\mathbb{R}} V - 2 = \dim X + 2 (\dim_{\mathbb{C}} V - 1)$; there is a fibration $\mathbb{P}V \to Z_{X,\rho} \to X$.
twistor_space.txt · Last modified: by spencer