We have thought a bit about the last paragraph of the above question and have some arguments as to what the answer should be. Since there have been no replies here so far, maybe I am allowed to hereby suggest an answer myself.
Recall, the last part of the above question was: is there a nonabelian 7dimensional ChernSimons theory holographically related to the nonabelian $(2,0)$theory on coincident M5branes, and if so, does it involve the Lagrangian that controls differential 5brane structures?
The following is an argument for the answer: Yes.
First, in Witten's AdS/CFT correspondence and TFT (hepth/9812012) a careful analysis of $AdS_5 /CFT_4$duality shows that the spaces of conformal blocks of the 4d CFT are to be identified with the spaces of states of (just) the ChernSimonstype Lagrangians inside the full type II action. At the very end of the article it is suggested that similarly the conformal blocks of the 6d $(2,0)$CFT are given by the spaces of states of (just) the ChernSimonspart inside 11d supergravity/Mtheory. But there only the abelian sugra effective Lagrangian
$$
\int_{AdS_7} \int_{S^4} C_3 \wedge G_4 \wedge G_4 = N \int_{AdS_7} C_3 \wedge G_4
$$
is briefly considered.
So we need to have a closer look at this: notice that there are two quantum corrections to the 11d sugra ChernSimons term.
First, the 11dimensional analog of the GreenSchwarz anomaly cancellation changes the above ChernSimons term to (from (3.14) in hepth/9506126 and ignoring prefactors here for notational simplicty)
$$
\int_{AdS_7} \int_{S^4} C_3 (\wedge G_4 \wedge G_4 + I_8(\omega))
=
N \int_{AdS_7} \left(
C_3 \wedge G_4

CS_7(\omega)
\right)
\,,
$$
for $I_8 = \frac{1}{48}(p_2  (\frac{1}{2}p_1)^2)$, where now the second term is the corresponding ChernSimons 7form evaluated in the spin connection (all locally).
So taking quantum anomaly cancellation into account, the argument of the above hepth/9812012 appears to predict a nonabelian 7d ChernSimons theory computing the conformal blocks of the 6d (2,0) theory, namely one whose field configurations involve both the abelian higher Cfield as well as the nonabelian spin connection field.
But there is a second quantum correction that further refines this statement: by Witten's On Flux Quantization In MTheory And The Effective Action (hepth/9609122) the underlying integral 4class $[G_4]$ of the $C$field in the 11d bulk is constrained to satisfy
$$
2[G_4] = \frac{1}{2}p_1  2a
\,,
$$
where on the right the first term is the fractional first Pontryagin class on $B Spin$ and where $a$ is the universal 4class of an $E_8$bundle, the one that in HoravaWitten compactification yields the $E_8$gauge field on the boundary of the 11d bulk. In that context, the boundary condition for the Cfield is $[G_4]_{bdr} = 0$, reducing the above condition to the 10d GreenSchwarz cancellation condition.
If this boundary condition on the $C$field is also relevant for the asymptotic $AdS_7$boundary, then this means that what locally lookes like a Spinconnection above is really a twisted differential String2connection with $2a$ being the twist. As discussed in detail there, such twisted differential String2connections involve a further field $H_3$ such that $d H_3 = tr(F_\omega \wedge F_\omega)  tr(F_{A_{E_8}} \wedge F_{A_{E_8}}))$. Plugging this condition into the above 7dimensional ChernSimons action adds to the abelian $C_3$field a ChernSimons term for the new $H_3$field, plus a bunch of nonabelian correction terms.
In total this argument produces a certain nonabelian 7d ChernSimons theory whose fields are twisted String2connections and whose states would yield the conformal blocks of a 6d CFT. Notice that by math/0504123 there is a gauge in which $String$2connections are given by loopgroup valued nonabelian 2forms (but there are other gauges in which this is not manifest). This is consistent with expectations for the "nonabelian gerbe theory" in 6d.
That's the physics argument, a more detailed writeup is in section 4.5.4.3.1 of my notes.
Now the point is this: in the next section, 4.5.4.3.2, it is shown that, independently of all of this physics handwaving, there is naturally a fully precise 7dimensional higher ChernSimons Lagrangian defined on the full moduli 2stack of twisted differential String2connections induced via higher ChernWeil theory from the second fractional Pontryagin class. As discussed there, on local differential form data this reproduces precisely the nonabelian 7d ChernSimons functional of the above argument.
We are in the process of writing this up as
Fiorenza, Sati, Schreiber, Nonabelian 7d ChernSimons theory and the 5brane . Comments are welcome.
This post imported from StackExchange Physics at 20140404 16:14 (UCT), posted by SEuser Urs Schreiber