2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/103605We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4k+1, generalizing positive energy representations of the loop group of the circle. We also classify such representations under a certain condition. The number of the equivalence classes of irreducible representations is finite, and is determined by the cohomology of the manifold.34 pages, LaTeX 2e; Lemma 6.4 is replaced, Subsection 7.2 is deletedRepresentation TheoryDifferential Geometrytext