Differential geometry of Grassmannian embeddings of based loop groups

In this article we study differential geometric properties of the most basic infinite-dimensional manifolds arising from fermionic (1 + 1)-dimensional quantum field theory: the restricted Grassmannian and the group of based loops in a compact simple Lie group. We determine the Riemann curvature tensor and the (linearly) divergent expression corresponding to the Ricci curvature of the restricted Grassmannian after proving that the latter manifold is an isotropy irreducible Hermitian symmetric space. Using the Gauss equation of the embedding of a based loop group into the restricted Grassmannian we show that the (conditional) Ricci curvature of a based loop group is proportional to its metric. Furthermore we explicitly derive the logarithmically divergent behaviour of several differential geometric quantities arising from this embedding.