Cech cocycles for differential characteristic classes – An -Lie theoretic construction
We define for every L-∞-algebra a smooth ∞-group integrating it, and define -principal ∞-bundles with connection. For every ∞-Lie algebra cocycle of suitable degree we give a refined ∞-Chern-Weil homomorphism that sends these -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 , 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 - 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 . This yields a differential refinement of the fractional second Pontryagin class – the Chern-Simons circle 7-bundle – which is not seen by the ordinary Chern-Weil homomorphism. We end by indicating how this serves to define differential string structures.
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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