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

2017/08/08 by Biswas, Indranil, Wong, Michael Lennox
#14M17 #32L10 #32L25 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1708.02658

Abstract

Given a holomorphic principal bundle Q \longrightarrow X, the universal space of holomorphic connections is a torsor C₁(Q) for ad Q ⊗ T^*X such that the pullback of Q to C₁(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 C₁(Q) may be identified with G/L, where L ⊂ P is a Levi factor. We use this identification to construct the twistor space associated to a natural hyper-Kähler metric on T^*(G/P), recovering Biquard's description of this twistor space, but employing only finite-dimensional, Lie-theoretic means.

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