Geometrical Methods in Quantum Field Theory: on the Topology of Spaces of Holomorphic Maps.

Available from UMI in association with The British Library. Requires signed TDF. In the thesis the space of based holomorphic maps, between a Riemann surface and a generalized flag manifold or a loop group, is compared with the corresponding space of continuous maps. Both spaces have connected components labeled by a (multi-) degree, and if the Riemann surface is the sphere, then it is shown that the two spaces have the same homology in the limit where the degree tends to infinity. The main idea is a generalization of the concept of a principal part of a meromorphic functions to a principal part of a holomorphic map into a flag manifold or a loop group. Then the space of holomorphic maps can be replaced with a configuration space of principal parts, which makes it possible to apply standard techniques in the proofs. The space of holomorphic maps from CP ^1 to a loop group Omega G can be identified with a moduli space of holomorphic G_{rm c}-bundles over CP^1 times CP^1. When CP^1 is replaced by a general Riemann surface X, then the space of holomorphic maps from X to Omega G is only a subset of the moduli space of bundles over X times CP^1 , but the space of configurations of principal parts can be identified with the full moduli space. Again it is shown that, when the degree tends to infinity, the homology of the moduli space tends to the homology of the space of continuous maps from X to Omega G.