Differential geometry of Grassmannian embeddings of based loop groups
Let \(F\) be a complex separable Hilbert space and a closed, infinite dimensional complex subspace \(F_{+} \) whose orthocomplement \(F_{-} \) is infinite dimensional as well. The restricted Grassmannian of the polarized Hilbert space \(F= F_{-}\oplus F_{+}\) is \(\text{Gr}(F,F_{+}) = { W \subset F |W}\) and is a closed complex subspace of \(F\) such that \(p_{+} : W \rightarrow F_{+}\) is Fredholm and \(p_{-}\) : \(W \rightarrow F_{-}\) is Hilbert-Schmidt. This is transitively acted upon by a ``restricted'' unitary group \(G\), and the isotropy group of the point \(F_{+}\) is the contractible group \(H= U(F_{+})\times U(F_{-})\). The authors prove that the reductive homogeneous space \(\text{Gr}(F,F_{+})=G/H\) is an isotropy irreducible Hermitian symmetric space. Then they compute the Riemannian curvature tensor of \(\text{Gr}(F,F_{+})\) as well as the ``linear divergence'' of its Ricci curvature tensor. For a compact Lie group \(K\) they consider the group \(LK\) of smooth loops in \(K\). There is an embedding \(LK/K \rightarrow \text{Gr}\) which was studied by means of the Gauss equations, and it is shown that the Ricci curvature of \(LK/K\) exists and is proportional to the metric induced from the Grassmannian. An Appendix on linear and logarithmic divergence is included.
scientific article; zbMATH DE number 1526472