Loop Groups, Higgs Fields and Generalised String Classes

Details Loop Groups, Higgs Fields and Generalised String Classes Loop Groups, Higgs Fields and Generalised String Classes

2009-06-26

arxiv.org/abs/0906.4843v1

Mathematics

Differential Geometry

PhD Thesis

Scientific paper

We consider various generalisations of the string class of a loop group bundle. The string class is the obstruction to lifting a bundle whose structure group is the loop group $LG$ to one whose structure group is the Kac-Moody central extension of the loop group. We develop a notion of higher string classes for bundles whose structure group is the group of based loops, $\Omega G$. In particular, we give a formula for characteristic classes in odd dimensions for such bundles which are associated to characteristic classes for $G$-bundles in the same way that the string class is related to the first Pontrjagyn class of a certain $G$-bundle associated to the loop group bundle in question. This provides us with a theory of characteristic classes for $\Omega G$-bundles analogous to Chern-Weil theory in finite dimensions. This also gives us a geometric interpretation of the well-known transgression map $H^{2k}(BG) \to H^{2k-1}(G).$ We also consider the obstruction to lifting a bundle whose structure group is not the loop group but the semi-direct product of the loop group with the circle, $LG \rtimes S^1$. We review the theory of bundle gerbes and their application to central extensions and lifting problems and use these methods to obtain an explicit expression for the de Rham representative of the obstruction to lifting such a bundle. We also relate this to a generalisation of the so-called `caloron correspondence' (which relates $LG$-bundles over $M$ to $G$-bundles over $M \times S^1$) to a correspondence which relates $LG \rtimes S^1$-bundles over $M$ to $G$-bundles over $S^1$-bundles over $M$.

Affiliated with

Also associated with

No associations

Rating

Loop Groups, Higgs Fields and Generalised String Classes does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Loop Groups, Higgs Fields and Generalised String Classes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Loop Groups, Higgs Fields and Generalised String Classes will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-670439