Positive Energy Representations of Gauge Groups I

Beschreibung

This is the first in a series of papers on projective positive energy representations of gauge groups. Let Ξ → M be a principal fiber bundle, and let Γc(M, Ad(Ξ)) be the group of compactly supported (local) gauge transformations. If P is a group of “space–time symmetries” acting on Ξ → M, then a projective unitary representation of Γc(M, Ad(Ξ)) ⋊ P is of positive energy if every “timelike generator” p0 ∈ 𝔭 gives rise to a Hamiltonian H(p0) whose spectrum is bounded from below. Our main result shows that in the absence of fixed points for the cone of timelike generators, the projective positive energy representations of the connected component Γc(M, Ad(Ξ))0 come from 1-dimensional P-orbits. For compact M this yields a complete classification of the projective positive energy representations in terms of lowest weight representations of affine Kac–Moody algebras. For noncompact M, it yields a classification under further restrictions on the space of ground states. In the second part of this series we consider larger groups of gauge transformations, which contain also global transformations. The present results are used to localize the positive energy representations at (conformal) infinity.

Autor*in

Bas Janssens
Delft University of Technology, The Netherlands

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infinite-dimensional Lie group unitary representation loop group gauge group positive energy representation projective representation

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