Models of higher Chern-Simons type

Question

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

Comments

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

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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

Answer 1

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

Comments

It is certainly fine to answer your own question if you later find the answer. It's certainly preferable to having unanswered questions.

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