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 $\simeq$ Top has a refinement to $\mathbf{H} =$ 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 $P$ to its corresponding Chern-Simons circle 7-bundle $\frac{1}{6}\mathbf{p}_2(P)$.
A choice of trivialization of $\frac{1}{6}p_2(P)$ is a fivebrane structure. The 6-groupoid of smooth fivebrane structures is the homotopy fiber of $\frac{1}{6}\mathbf{p}_2$ over the trivial circle 7-bundle.
By Chern-Weil theory in Smooth∞Grpd this morphism may be further refined to a differential characteristic class $\frac{1}{6}\hat \mathbf{p}_2$ that lands in the ordinary differential cohomology $\mathbf{H}_{diff}(X, \mathbf{B}^7 U(1))$, classifying circle 7-bundles with connection
The 7-groupoid of differential fivebrane structures is the homotopy fiber of this refinement $\frac{1}{6}\hat \mathbf{p}_2$ 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 $X$ is characterized by a tuple consisting of
a 2-connection $\nabla$ on a String-principal 2-bundle on $X$;
a choice of trivial circle 7-bundle with connection $(0, H_7)$, hence a differential 7-form $H_7 \in \Omega^7(X)$;
a choice of equivalence $\lambda$ of the Chern-Simons circle 7-bundle with connection $\frac{1}{6}\hat\mathbf{p}_2(\nabla)$ of $\nabla$ with this chosen 7-bundle
More generally, one can consider the homotopy fibers of $\frac{1}{6}\hat \mathbf{p}_2$ over arbitrary circle 7-bundles with connection $\hat \mathcal{G}_8 \in \mathbf{H}_{diff}^8(X, \mathbf{B}^3 U(1))$ and hence replace $(0,H_7)$ in the above with $\hat \mathcal{G}_8$. According to the general notion of twisted cohomology, these may be thought of as twisted differential string structures, where the class $[\mathcal{G}_8] \in H^8_{diff}(X)$ is the twist.
We will assume that the reader is familiar with basics of the discussion at Smooth∞Grpd. We often write $\mathbf{H} := Smooth \infty Grpd$ for short.
Let $String(n) \in$ Smooth∞Grpd be the smooth String 2-group, for some $n \gt 6$,^{1} regarded as a Lie 2-group and thus canonically as an ∞-group object in Smooth∞Grpd. We shall notationally suppress the $n$ in the following. Write $\mathbf{B}String$ for its delooping of $Spin$ in Smooth∞Grpd. (See the discussion here). Let moreover $\mathbf{B}^6 U(1) \in Smooth \infty Grpd$ be the circle Lie 7-group and $\mathbf{B}^7 U(1)$ 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 $\mathfrak{so}$ 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 $\Pi : Smooth \infty Grpd \to$ ∞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 $\langle - , - ,-,-\rangle$ on $\mathfrak{so}$.
For any $X \in$ Smooth∞Grpd, the 6-groupoid of differential fivebrane-structures on $X$ – $Fivebrane_{diff}(X)$ – is the homotopy fiber of $\frac{1}{6}\hat \mathbf{p}_2(X)$ over the trivial differential cocycle.
More generally (see twisted cohomology) the 6-groupoid of twisted differential fivebrane structures is the (∞,1)-pullback $Fivebrane_{diff,tw}(X)$ in
where the right vertical morphism is a choice of (any) one point in each connected component (differential cohomology class) of the cocycle ∞-groupoid $\mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))$ (the homotopy type of the (∞,1)-pullback is independent of this choice).
In terms of local ∞-Lie algebra valued differential forms data this has been considered in (SSSIII), as we shall discuss below.
We use the presentation of the (∞,1)-topos Smooth∞Grpd (as described there) by the local model structure on simplicial presheaves $[CartSp_{smooth}^{op}, sSet]_{proj,loc}$ 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 $\frac{1}{6} \hat \mathbf{p}_2$ is presented in $[CartSp_{smooth}^{op}, sSet]_{proj}$ 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 $\frac{1}{6}\hat \mathbf{p}_2$ we now find a resolution of this morphism $\exp(\mu,cs)$ by a fibration in $[CartSp_{smooth}^{op}, sSet]_{proj}$. By the fact that this is a simplicial model category then also the hom of any cofibrant object into this morphism, computing the cocycle $\infty$-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.
(…)
The discussion is analogous to that at differential string structure. To be filled in
(…)
Whitehead tower of the orthogonal group
twisted differential c-structure
differential fivebrane structure
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.
for $n=3,4,5,6$ $String(n)$ is not 6-connected, so one must take extra care with the intervening stages in the Whitehead tower. ↩