Higher Holonomies, Geometric Loop Groups and Smooth Deligne Cohomology

Abstract

A holonomy is a structure associated with any connection $\nabla$ on a smooth principal G-bundle over a smooth manifoldM.It assigns to every piecewise smooth loop $\gamma :{{S}^{1}} \to M$ an element ${{h}^{\nabla }}(\gamma )$ ofG.If$\nabla$is flat, then${h^\nabla }(\sigma )$depends only on the relative homotopy class ofaand hence, it induces a homomorphism${h^\nabla }:{\pi _1}(M) \to G$It is well-known that the assignment$\nabla \mapsto {h^\nabla }$induces an isomorphism

$$h:L\left( {M,G,{{\nabla }^{{flat}}}} \right) \to Hom\left( {{{\pi }_{1}}\left( M \right),G} \right), $$

(1)

where$L(M,G,{\nabla ^{flat}})$is the pointed set of isomorphism classes of smooth principal G-bundles with a flat connection overM.In this paper we investigate several generalizations of the isomorphism (1).