Critical string models
Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
Where a fivebrane structure is a trivialization of a class in integral cohomology, a differential fivebrane structure is the trivialization of this class refined to ordinary differential cohomology:
the second fractional Pontryagin class
in the (∞,1)-topos ∞Grpd Top has a refinement to Smooth∞Grpd of the form
– the smooth second fractional Pontryagin class.
The induced morphism on cocycle ∞-groupoids
sends a string 2-group-principal 2-bundle to its corresponding Chern-Simons circle 7-bundle .
A choice of trivialization of is a fivebrane structure. The 6-groupoid of smooth fivebrane structures is the homotopy fiber of over the trivial circle 7-bundle.
By Chern-Weil theory in Smooth∞Grpd this morphism may be further refined to a differential characteristic class that lands in the ordinary differential cohomology , classifying circle 7-bundles with connection
The 7-groupoid of differential fivebrane structures is the homotopy fiber of this refinement over the trivial circle 7-bundle with trivial connection or more generally over the trivial circle 7-bundles with possibly non-trivial connection.
Such a differential fivebrane structure over a smooth manifold is characterized by a tuple consisting of
a 2-connection on a String-principal 2-bundle on ;
a choice of trivial circle 7-bundle with connection , hence a differential 7-form ;
a choice of equivalence of the Chern-Simons circle 7-bundle with connection of with this chosen 7-bundle
More generally, one can consider the homotopy fibers of over arbitrary circle 7-bundles with connection and hence replace in the above with . According to the general notion of twisted cohomology, these may be thought of as twisted differential string structures, where the class is the twist.
We will assume that the reader is familiar with basics of the discussion at Smooth∞Grpd. We often write for short.
Let Smooth∞Grpd be the smooth String 2-group, for some ,1 regarded as a Lie 2-group and thus canonically as an ∞-group object in Smooth∞Grpd. We shall notationally suppress the in the following. Write for its delooping of in Smooth∞Grpd. (See the discussion here). Let moreover be the circle Lie 7-group and its delooping.
At Chern-Weil theory in Smooth∞Grpd the following statement is proven (FSS).
The image under Lie integration of the canonical Lie algebra 7-cocycle
on the semisimple Lie algebra of the Spin group – the special orthogonal Lie algebra – is a morphism in Smooth∞Grpd of the form
whose image under the the fundamental ∞-groupoid (∞,1)-functor/ geometric realization ∞Grpd is the ordinary second fractional Pontryagin class
in Top. Moreover, the corresponding refined differential characteristic class
is in cohomology the corresponding refined Chern-Weil homomorphism
with values in ordinary differential cohomology that corresponds to the second Killing form invariant polynomial on .
For any Smooth∞Grpd, the 6-groupoid of differential fivebrane-structures on – – is the homotopy fiber of over the trivial differential cocycle.
More generally (see twisted cohomology) the 6-groupoid of twisted differential fivebrane structures is the (∞,1)-pullback in
where the right vertical morphism is a choice of (any) one point in each connected component (differential cohomology class) of the cocycle ∞-groupoid (the homotopy type of the (∞,1)-pullback is independent of this choice).
Construction in terms of -Cech cocycles
We use the presentation of the (∞,1)-topos Smooth∞Grpd (as described there) by the local model structure on simplicial presheaves to give an explicit construction of twisted differential fivebrane structures in terms of Cech-cocycles with coefficients in ∞-Lie algebra valued differential forms.
Recall the following fact from Chern-Weil theory in Smooth∞Grpd (FSS).
The differential second fractional Pontryagin class is presented in by the top morphism of simplicial presheaves in
Here the middle morphism is the direct Lie integration of the L-∞ algebra cocycle while the top morphisms is its restriction to coefficients for ∞-connections.
In order to compute the homotopy fibers of we now find a resolution of this morphism by a fibration in . By the fact that this is a simplicial model category then also the hom of any cofibrant object into this morphism, computing the cocycle -groupoids, is a fibration, and therefore, by the general discussion at homotopy pullback, we obtain the homotopy fibers as the ordinary fibers of this fibration.
Presentation of the differential class by a fibration
The discussion is analogous to that at differential string structure. To be filled in
Whitehead tower of the orthogonal group
The local data for the ∞-Lie algebra valued differential forms for the description of twisted differential fivebrane structures as above was given in
The full Cech-Deligne cocycles induced by this and their homotopy fibers were discussed in
A general discussion is at
in section 4.2.