# Models of higher Chern-Simons type

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It has long been clear that (the action functional of) Chern-Simons theory has various higher analogs and variations of interest. This includes of course traditional higher dimensional Chern-Simons theory (abelian and non-abelian) as well as the algebroid-version: the Courant-sigma model, but also seemingly more remote systems such as string field theory (and hence essentialy also its effective truncations), a fact that is already somewhat remarkable.

In a recent article we claimed that there is a systematic sense in which also all AKSZ sigma-models are special cases of a general abstract notion of "infinity-Chern-Simons theory". These AKSZ models include, in turn, also the Poisson sigma-model (hence also the A-model and the B-model). Also BF-theory coupled to topological Yang-Mills theory fits in.

Therefore in a precise sense all these systems are examples of a single underlying basic mechanism. My question is: can you point out other models of interest in the literature (or in your drawer) that look like they might be "of generalized Chern-Simons type", along these lines? (I am not just looking, say, for "Chern-Simons term"-summands in higher supergravity actions, even though these are related, but for example new variants of full higher dimensional Chern-Simons (super)gravity.)

For instance: has there been a proposal for a nonabelian 7-dimensional Chern-Simons-type model that might be the holographic partner to the self-dual nonabelian 6d (2,0)-superconformal QFT (so that the state spaces of the former are the conformal blocks of the latter)? While we did come across a natural non-abelian 7-dimensional Chern-Simons type TQFT whose fields are string-2-connections (here), I am not sure how to see if this might be the relevant one. Do you?

This post imported from StackExchange Physics at 2014-04-04 16:14 (UCT), posted by SE-user Urs Schreiber
I'm confused -- the 6d (2,0) theories are dual to M-theory on AdS$_7 \times S^4$. Doesn't that preclude them being dual to a Chern-Simons theory?

This post imported from StackExchange Physics at 2014-04-04 16:14 (UCT), posted by SE-user Matt Reece
Thanks for the comment. Let's see, maybe I mean just the 2-form part of it. I was reasoning like this, please let me know if this makes sense to you: in the abelian situation the 6d theory is supposed to contain a self-dual 2-form field. These are known (sometimes even defined this way) as the boundary theory of 7d abelian CS theory, as in hep-th/9610234. I gather the search is on for the nonabelian generalization of the self-dual 2-form theory, hence it seems we should also expect there to be a corresponding nonabelian generalization of that 7d CS theory.

This post imported from StackExchange Physics at 2014-04-04 16:14 (UCT), posted by SE-user Urs Schreiber

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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
answered Oct 12, 2011 by (6,095 points)
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