Positive Energy Representations of Gauge Groups

Abstract

Recent progress on the representation theory of certain infinite dimensional gauge groups has raised an interest in the strongly continuous unitary representations of groups of a specific form that satisfy a certain positive energy condition. An equivalent formulation of the positive energy condition is obtained, allowing for a geometrical interpretation of this condition and which yields necessary conditions for satisfying this condition. By the theory of the Mackey machine, the strongly continuous unitary representations of such groups that are of positive energy are classified by corresponding stabilizer subgroups. In a specific case, these are fully determined up to equivalence.\\ Finally, a method is developed that embeds homogeneous bundles as eigenspace subbundles of trivial bundles that in particular applies to the bundles obtained through the representation theory of groups mentioned above. The eigenspace subbundles thus obtained allow for a more detailed understanding of the induced representation and moreover resemble various theories in particle physics.