Domenico Fiorenza, Urs Schreiber, Jim Stasheff,
Cech cocycles for differential characteristic classes β An $\infty$-Lie theoretic construction
We define for every L-β-algebra $\mathfrak{g}$ a smooth β-group $G$ integrating it, and define $G$-principal β-bundles with connection. For every β-Lie algebra cocycle of suitable degree we give a refined β-Chern-Weil homomorphism that sends these $\infty$-bundles to classes in ordinary differential cohomology that lift the corresponding curvature characteristic forms.
As a first example we show that applied to the canonical 3-cocycle on a semisimple Lie algebra $\mathfrak{g}$, this construction reproduces the Cech-Deligne cocycle representative for the first differential Pontryagin class that was found by Brylinski-MacLaughlin. This is the Chern-Simons circle 3-bundle with connection. If its class vanishes there is a lift to a String 2-group-2-connection on a smooth $String(G)$- principal 2-bundle. As a second example we describe the higher Chern-Weil-homomorphism applied to this String-bundle which is induced by the canonical degree 7-cocycle on $\mathfrak{g}$. This yields a differential refinement of the fractional second Pontryagin class β the Chern-Simons circle 7-bundle β which is not seen by the ordinary Chern-Weil homomorphism. We end by indicating how this serves to define differential string structures.
In
this is section 3.3.13 (general theory) and sections 4.1 and 4.4 (application to String- and Fivebrane structures).
A survey talk with some related material is at
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