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November 7, 2010

Cocycles for Differential Characteristic Classes

Posted by Urs Schreiber

The previous entry mentioned that Chern-Weil theory exists in every cohesive \infty-topos H\mathbf{H}. For H=LieGrpd\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 L_\infty-algebra 𝔤\mathfrak{g} a smooth \infty-group GG integrating it, and define GG-principal \infty-bundles with connection. For every L 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 𝔤\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 String(G)\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 𝔤\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.

Posted at November 7, 2010 7:10 PM UTC

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