Infinite-Dimensional Lie Theory for Gauge Groups

• Main Author: Wockel, Christoph
• Format: Others
• Language: English; en
• Published: 2006
• Online Access: http://tuprints.ulb.tu-darmstadt.de/752/1/diss.pdf; Wockel, Christoph <http://tuprints.ulb.tu-darmstadt.de/view/person/Wockel=3AChristoph=3A=3A.html> : Infinite-Dimensional Lie Theory for Gauge Groups. [Online-Edition] Technische Universität, Darmstadt [Ph.D. Thesis], (2006)

The aim of this thesis is to consider symmetry groups of principal bundles and to initiate a Lie theoretic treatment of these groups. These groups of main interest are called gauge groups. When taking a particular principal K-bundle P into account, we denote the gauge group of this bundle by Gau(P), which we mostly identify with the space of smooth K-equivariant mappings C∞(P,K)K. These groups will be treated as infinite-dimensional Lie groups, modelled on an appropriate vector space. Since Lie theory in infinite dimensions is a research area which is presently under active development, this terminology is not settled, and we have to make precise what we mean with ``infinite-dimensional Lie theory''. The following questions are considered in this thesis: For which bundles P is Gau(P) an infinite-dimensional Lie group, modelled on an appropriate locally convex space? How can the homotopy groups πn(Gau(P)) be computed? What extensions does Gau(P) permit? Of course, this is only a marginal part of the questions that come along with Lie groups. These problems have in common that they can be approached with the same idea, which we describe now. Along with a bundle P come many different ways of describing it (up to equivalence). Two fundamental different ways are given by describing P either in terms of a classifying map fP, or by a cocycle kP. A classifying map fP is a globally defined map fP with values in some classifying space, while a cocycle consists of many locally defined maps, with values in a Lie group, obeying some compatibility conditions. These objects, classifying maps and cocycles, live in two different worlds, namely topology and Lie theory. The idea now is to combine these two concepts and to use the existing tools from topology and Lie theory in order to give answers to the questions above. Since the questions are formulated quite generally, we cannot hope to get answers in full generality, but for many interesting cases occurring in mathematical physics, we will provide answers. These include: Construction of a Lie group structure on Gau(P) if the structure group is locally exponential. Showing that the canonical inclusion Gauc(P)→ Gau(P) is a weak homotopy equivalence. Providing a smoothing procedure for continuous principal bundles. Construction of an Extension of Lie groups Gau(P)→Aut(P)→ Diff(M)P. Calculation of some homotopy groups and of all rational homotopy groups of Gau(P) for finite-dimensional principal bundles over spheres. Construction of central extensions Z→ GP → Gau(P)0. Construction of an automorphic action of Aut(P) on GP. Applications to affine twisted Kac-Moody groups.