The universal connection for principal bundles over homogeneous spaces and twistor space of coadjoint orbits

TL;DR

This paper explores the universal space of holomorphic connections on principal bundles over homogeneous spaces, identifying it with a quotient of a complex Lie group, and constructs the associated twistor space using Lie-theoretic methods.

Contribution

It provides a Lie-theoretic construction of the twistor space of coadjoint orbits, connecting universal connection spaces with homogeneous space structures.

Findings

Identifies the universal connection space with a quotient G/L for certain bundles.

Constructs the twistor space of hyper-Kähler metrics on cotangent bundles of homogeneous spaces.

Recovers Biquard's twistor space description using finite-dimensional Lie theory.

Abstract

Given a holomorphic principal bundle $Q\, \longrightarrow\, X$, the universal space of holomorphic connections is a torsor $C_1(Q)$ for $\text{ad} Q \otimes T^*X$ such that the pullback of $Q$ to $C_1(Q)$ has a tautological holomorphic connection. When $X\,=\, G/P$, where $P$ is a parabolic subgroup of a complex simple group $G$, and $Q$ is the frame bundle of an ample line bundle, we show that $C_1(Q)$ may be identified with $G/L$, where $L\, \subset\, P$ is a Levi factor. We use this identification to construct the twistor space associated to a natural hyper-K\"ahler metric on $T^*(G/P)$, recovering Biquard's description of this twistor space, but employing only finite-dimensional, Lie-theoretic means.