The n-Category Café

Cocycles for Differential Characteristic Classes

Posted by Urs Schreiber

The previous entry mentioned that Chern-Weil theory exists in every cohesive $\infty$-topos $\mathbf{H}$. For $\mathbf{H} = \infty LieGrpd$ the topos of smooth $\infty$-groupoids, this reproduces ordinary Chern-Weil theory – and generalizes it from smooth principal bundles over Lie groups to principal $\infty$-bundles over Lie $\infty$-groups.

Some basics of this $\infty$-Chern-Weil theory in the smooth context we have been trying to write up a bit more. Presently the result is this

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff,

  Cocycles for differential characteristic classes - An $\infty$-Lie theoretic construction

  (pdf)

Abstract We define for every $L_\infty$-algebra $\mathfrak{g}$ a smooth $\infty$-group $G$ integrating it, and define $G$-principal $\infty$-bundles with connection. For every $L_\infty$-algebra coycle of suitable degree we give a refined $\infty$-Chern-Weil homomorphism that sends these $\infty$-bundles to classes in differential cohomology that lift the corresponding curvature characteristic classes.

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. If its class vanishes there is a lift to a $\mathrm{String}(G)$-connection on a smooth String-2-group principal 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 which is not seen by the ordinary Chern-Weil homomorphism. We end by indicating how this serves to define differential String-structures.