A Chern-Simons circle 7-bundle is the circle 7-bundle with connection classified by the cocycle in degree-8 ordinary differential cohomology that is canonically associated to a string group-principal 2-bundle with connection.
The characteristic class called the second fractional Pontryagin class $\frac{1}{6}p_2 : \mathcal{B}String \to \mathcal{B}^8 \mathbb{Z}$ in Top on the classifying space of the string group has a smooth lift to the smooth second fractional Pontryagin class
in $\mathbf{H} :=$ ∞LieGrpd, mapping from the delooping ∞-Lie groupoid of the string Lie 2-group to that of the circle Lie 7-group. This is the Lie integration of the degree 7 ∞-Lie algebra cocycle $\mu_7 : \mathfrak{string} \to b^6 \mathbb{R}$ on the string Lie 2-algebra which classified the fivebrane Lie 6-algebra.
Therefore, by ∞-Chern-Weil theory, there is a refinement of this morphism to ∞-bundles with connection
hence on cocycle ∞-groupoids
a map from string Lie 2-group-principal 2-bundles with connection to circle 7-bundles with connection, hence degree 8 ordinary differential cohomology.
For $(P,\nabla)$ a String-principal 2-bundle, we call the image $\frac{1}{6}\hat\mathbf{p}(\nabla) \in \mathbf{H}_{diff}(X,\mathbf{B}^z U(1))$ its Chern-Simons circle 7-bundle with connection.
This is a differential refinement of the obstruction to lifting $P$ to a fivebrane Lie 6-group-bundle.
By construction, the curvature 8-form of $\hat \mathbf{c}(\nabla)$ is the curvature characteristic form $\langle F_\nabla \wedge F_\nabla \wedge F_\nabla \wedge F_\nabla\rangle$ of $\nabla$ and accordingly the 7-form connection on $\hat \mathbf{c}(\nabla)$ is locally a Chern-Simons form $CS(\nabla)$ of $\nabla$.
Therefore the higher parallel transport induced by $\frac{1}{6}\hat \mathbf{p}_2(\nabla)$ over 7-dimensional volumes $\phi : \Sigma \to X$ is the action functional of degree-7 ∞-Chern-Simons theory. This is the analog of the way the Chern-Simons circle 3-bundle arises from Spin-principal bundles.
Using the discusson at ∞-Chern-Weil theory and in direct analogy to the constructin of the Chern-Simons circle 3-bundle we can model the (∞,1)-functor
by postcomposition with the ∞-anafunctor
where $\mu_7 : \mathfrak{string} \to b^6 \mathbb{R}$ is the 7-cocycle that classifies the fivebrane Lie 6-algebra.
For
an ∞-anafunctor modelling a cocycle for a string 2-group-principal 2-bundle with connection on a 2-bundle the $\infty$-anafunctor composition
produces a lift of the transition functions $g$ to $\mathbf{cosk}_7 \exp(\mathfrak{string})$. The string-cocycle is itself in first degree a collection of paths in $G$, in second a collection of surfaces with labels in $U(1)$. That lift corresponds to further resolving this to families
up to $k = 7$. That this is indeed always possible is the statement about Lie integration that $\mathbf{cosk}_7 \exp(\mathfrak{string}) \stackrel{\simeq}{\to} \mathbf{B}String$ is a weak equivalence, which in turn is due to the fact that the next nonvanishing homotopy group of $G = SO(n)$ after $\pi_3$ is $\pi_7$.
The above composite ∞-anafunctor is manifestly a degree 8-cocycle in Cech-Deligne cohomology given by
where $A$ is a connection form on the total space of the $Spin(n)$-principal bundle that the string bundle itself is lifted from and $CS_7$ is the Chern-Simons element in degree 7 defining the fivebrane Lie 6-algebra.
(…)
universal Chern-Simons n-bundle
Chern-Simons circle 7-bundle
The CS 7-bundle serves as the extended Lagrangian for a 7d Chern-Simons theory. See there for more.
The CS 7-bundle as an circle 7-bundle with connection on the smooth moduli infinity-stack of string 2-group-2-connections has been constructed in
and identified as part of the 11-dimensional supergravity Chern-Simons terms after KK-reduction on $S^4$ to 7-dimensional supergravity (for AdS/CFT duality with the M5-brane worldvolume 6d (2,0)-superfonformal ∞-Wess-Zumino-Witten theory) in
Domenico Fiorenza, Hisham Sati, Urs Schreiber, 7d Chern-Simons theory and the 5-brane
Domenico Fiorenza, Hisham Sati, Urs Schreiber, The moduli 3-stack of the C-field