Projective unitary representations of smooth Deligne cohomology groups

TL;DR

This paper constructs and classifies projective unitary representations of smooth Deligne cohomology groups for certain manifolds, extending concepts from loop group representations and linking the classification to the manifold's cohomology.

Contribution

It introduces a method to construct and classify projective unitary representations of smooth Deligne cohomology groups on high-dimensional manifolds, generalizing known loop group results.

Findings

Finite number of irreducible representation classes determined by manifold cohomology

Construction of projective unitary representations for manifolds of dimension 4k+1

Classification under specific conditions

Abstract

We 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.