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 7-dimensional Chern-Simons theory holographically related to the nonabelian $(2,0)$-theory on coincident M5-branes, and if so, does it involve the Lagrangian that controls differential 5-brane structures?
The following is an argument for the answer: Yes.
First, in Witten's AdS/CFT correspondence and TFT (hep-th/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 Chern-Simons-type 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 Chern-Simons-part inside 11d supergravity/M-theory. 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 Chern-Simons term.
First, the 11-dimensional analog of the Green-Schwarz anomaly cancellation changes the above Chern-Simons term to (from (3.14) in hep-th/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 Chern-Simons 7-form evaluated in the spin connection (all locally).
So taking quantum anomaly cancellation into account, the argument of the above hep-th/9812012 appears to predict a non-abelian 7d Chern-Simons theory computing the conformal blocks of the 6d (2,0) theory, namely one whose field configurations involve both the abelian higher C-field as well as the non-abelian spin connection field.
But there is a second quantum correction that further refines this statement: by Witten's On Flux Quantization In M-Theory And The Effective Action (hep-th/9609122) the underlying integral 4-class $[G_4]$ 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 4-class of an $E_8$-bundle, the one that in Horava-Witten compactification yields the $E_8$-gauge field on the boundary of the 11d bulk. In that context, the boundary condition for the C-field is $[G_4]_{bdr} = 0$, reducing the above condition to the 10d Green-Schwarz 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 Spin-connection above is really a twisted differential String-2-connection with $2a$ being the twist. As discussed in detail there, such twisted differential String-2-connections 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 7-dimensional Chern-Simons action adds to the abelian $C_3$-field a Chern-Simons term for the new $H_3$-field, plus a bunch of nonabelian correction terms.
In total this argument produces a certain nonabelian 7d Chern-Simons theory whose fields are twisted String-2-connections and whose states would yield the conformal blocks of a 6d CFT. Notice that by math/0504123 there is a gauge in which $String$-2-connections are given by loop-group valued nonabelian 2-forms (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 7-dimensional higher Chern-Simons Lagrangian defined on the full moduli 2-stack of twisted differential String-2-connections induced via higher Chern-Weil theory from the second fractional Pontryagin class. As discussed there, on local differential form data this reproduces precisely the nonabelian 7d Chern-Simons functional of the above argument.
We are in the process of writing this up as
Fiorenza, Sati, Schreiber, Nonabelian 7d Chern-Simons theory and the 5-brane . Comments are welcome.
This post imported from StackExchange Physics at 2014-04-04 16:14 (UCT), posted by SE-user Urs Schreiber
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