Canonical Fibrations on Geodesics

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There is a realisation of the canonical fibrations of flag manifolds that serves to introduce a twistor space. For this, assume that G is of adjoint type (i.e. has trivial centre) and let ΩG denote the infinite-dimensional manifold of based loops in G: the loop group. In fact ΩG is a Kähler manifold and may be viewed as a flag manifold GC/P where GC is the manifold of loops in GC and P is the subgroup of those that extend holomorphically to the disc. We have various fibrations ρλ: ΩG → G given by evaluation at λ ∈ S1 and in some ways ρ−1 behaves like a canonical fibration making ΩG into a universal twistor space for G. It is a theorem of Uhlenbeck that any harmonic map of S2 into G is of the form ρ−1 ◦ Φ for some “super-horzontal” holomorphic map Φ : S2 → ΩG.

The flag manifolds of G embed in ΩG as conjugacy classes of geodesics and we find a particular embedding of this kind using the canonical element. Indeed, our assumption that G be centre-free means that exp 2πξ = e for any canonical element ξ. Thus if F = G/H = GC/P is a flag manifold with ξ the canonical element of p, we may define a map Γ: F → ΩG by setting

Γ(eH) = (e√−1t → exp tξ)

and extending by equivariance. Moreover, if N is the inner symmetric space associated to F, we have a totally geodesic immersion γ : N → G defined by setting γ(x) equal to the element of G that generates the involution at x. We now have:

Γ: F → ΩG is a totally geodesic, holomorphic, isometric immersion and the following diagram commutes

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where π1 is a canonical fibration. Thus we have a realisation of the canonical fibrations as the trace of ρ−1 on certain conjugacy classes of geodesics.

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