January 21, 2012

Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

Posted by Urs Schreiber

We are in the process of finalizing a little article:

Domenico Fiorenza, Hisham Sati, Urs Schreiber,
Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

Below is the abstract, and, below the fold, the beginning of the introduction. A pdf with the current version is behind the above link.

We would be grateful for comments.

The article is written in a non-formal style with an audience of certain physicists in mind, but ample pointers are given to places where all the ingredients are spelled out in detail and precisely.

Abstract

The worldvolume theory of coincident M5-branes is expected to contain a nonabelian 2-form/nonabelian gerbe gauge theory that is a higher analog of self-dual Yang-Mills theory. But the precise details – in particular the global moduli / instanton / magnetic charge structure – have remained elusive.

Here we argue that the holographic dual of this nonabelian 2-form field, under $\mathrm{AdS}_7/\mathrm{CFT}_6$ duality, can be deduced from anomaly cancellation.

We find this way a 7-dimensional nonabelian Chern-Simons theory of twisted String 2-connection fields, which, in a certain higher gauge, are given locally by non-abelian 2-forms with values in a Kac-Moody loop Lie algebra. We construct the corresponding action functional on the entire smooth moduli 2-stack of field configurations, thereby defining the theory globally, at all levels and with the full instanton structure, which is nontrivial due to the twists imposed by the quantum corrections. Along the way we explain some general phenomena of higher nonabelian gauge theory that we need.

Introduction

The quantum field theory (QFT) on the worldvolume of M5-branes is known to be a 6-dimensional (2,0)-superconformal theory that contains a 2-form potential field $B_2$, whose 3-form field strength $H_3$ is self-dual. Whatever it is precisely and in generality, this QFT has been argued to be the source of deep physical and mathematical phenomena, such as Montonen-Olive S-duality, geometric Langlands duality, and Khovanov homology. Yet, and despite this interest, a complete description of the precise details of this QFT is still lacking.

In particular, as soon as one considers the worldvolume theory of several coincident M5-branes, the 2-form appearing locally in this 6d QFT is expected to be nonabelian (to take values in a nonabelian Lie algebra). But a description of this nonabelian gerbe theory has been elusive. Here we present a consistent formulation of nonabelian 2-form fields and propose dynamics for them under holography.

On the other hand, for a single M5-brane the Lagrangian of the theory has been formulated. Furthermore, in this abelian case there is a holographic dual description of the 6d theory by 7-dimensional abelian Chern-Simons theory, as part of $\mathrm{AdS}_7/\mathrm{CFT}_6$-duality.

We give here an argument, following Witten96, Witten98 but taking the quantum anomaly cancellation of the M5-brane in 11-dimensional supergravity into account, that in the general case the $\mathrm{AdS}_7$/$\mathrm{CFT}_6$-duality involves a 7-dimensional nonabelian Chern-Simons action that is evaluated on higher nonabelian gauge fields which we identify as _twisted 2-connections over the String-2-group.

Then we give a precise description of a certain canonically existing 7-dimensional nonabelian gerbe-theory on boundary values of quantum-corrected supergravity field configurations in terms of nonabelian differential cohomology. We show that this has the properties expected from the quantum anomaly structure of 11-dimensional supergravity. In particular, we discuss that there is a higher gauge in which these field configurations locally involve non-abelian 2-forms with values in the Kac-Moody central extension of the loop Lie algebra of the special orthogonal Lie algebra $\mathfrak{so}$ and of the exceptional Lie algebra of E8. We also describe the global structure of the smooth moduli 2-stack of field configurations, which is more subtle.

Posted at January 21, 2012 11:13 PM UTC

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Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

Does any extra light get shone on geometric Langlands duality and Khovanov homology from your paper?

Typos:

p. 48 different but but equivalent

p. 50 The degeewise ordinary

Posted by: David Corfield on January 22, 2012 4:37 PM | Permalink | Reply to this

Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

Hi David,

thanks for catching these typos. A new version with these and many other typos corrected is now uploaded.

The question you ask is of course very intersting, but also hard. We have now edited the opening paragraph such as not to make the impression as if the article is meant to answer this question, motivating as it may be, and to amplify that the article is concerned with the more modest intermediate goal of identifying a consistent nonabelian 2-form field kinematics and a dual dynamics.

The next step in this direction, that we are working on, is discussing the actual duality step that goes from the states of the 7d theory to the conformal blocks of the 6d theory. A while ago I had a blog post here on how every $\infty$-Chern-Simons theory comes with its “WZW model” in one dimension lower. Here we get a 6-dimensional WZW model whose target space is the (group-)stack $\mathrm{String}$ (or rather a differential twist thereof). That is to be analyzed.

For the moment we just wished to mention that all this effort should be well motivated in view of the fact that the system it is concerned with is known to also be at the heart of a bunch of other interesting phenomena, both of physical as well as of pure mathematical interest.

Posted by: Urs Schreiber on January 23, 2012 10:01 AM | Permalink | Reply to this

Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

I am working on a little essay,

Posted by: Urs Schreiber on January 23, 2012 11:32 PM | Permalink | Reply to this

Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

That’s clearly expressed. Are you thinking of publishing the essay elsewhere? What signs are there that people are getting the message?

Posted by: David Corfield on January 24, 2012 9:24 AM | Permalink | Reply to this

Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

That’s clearly expressed.

Thanks for the feedback. That’s useful to know. I am still cycling a bit through fine-tuning it, adding more aspects, and then, inevitably, having to fine-tune again.

Are you thinking of publishing the essay elsewhere?

I haven’t thought yet of formally publishing it anywhere, no. Maybe worth a thought if there is a suitable occasion.

What signs are there that people are getting the message?

I guess the answer to that is: it could be better, but progress is visible. We are working on it.

Posted by: Urs Schreiber on January 24, 2012 10:49 AM | Permalink | Reply to this

Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

What happens to the differential cohomological structures as you pass from one cohesive topos to another?

If we are allowed some outlandish speculation, is it possible that the commonality being found between geometric Langlands and number theoretic Langlands – e.g., use of Hitchin fibrations in the Fundamental Lemma, etc. – comes about because there are arithmetic forms of cohesiveness (such as your question ask for at MO, so that one should expect machinery that works in the case of smooth cohesion to carry over to arithmetic situations?

If the remark of Laumon here

Natural expectation: The Fundamental Lemma is the consequence of a (stronger) cohomological statement

is correct, and you are correct that behind any cohomology there is a hom-space in an $(\infty, 1)$-topos, that would be the beginning of an answer.

Posted by: David Corfield on January 24, 2012 11:56 AM | Permalink | Reply to this

Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

In this generality, what does cohoomology’ mean?
For topological spaces, we have axioms for a cohomology theory’.

Posted by: jim stasheff on January 24, 2012 2:16 PM | Permalink | Reply to this

Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

There’s a slogan from the nLab cohomology page

Thousands of definitions of notions of cohomology and its variants. From the nPOV, just a single concept: an $\infty$-categorical hom-space in an $(\infty,1)$-topos.

Posted by: David Corfield on January 24, 2012 3:19 PM | Permalink | Reply to this

Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

In this generality, what does cohoomology’ mean?

Over a paracompact manifold it means nonabelian Čech hypercohomology. Over an orbifold or moduli stack or whatever, it means the correct equivariant generalization of that.

This is explained in section 2.1.6 of DCCT.

For topological spaces, we have axioms for a cohomology theory’.

And then the Brown representability theorem shows that these axioms equivalently characterize homotopy classes of maps into some stable space. If the space is not stable, people call this nonabelian cohomology. If instead the stable space is replaced by a “stable sheaf”, they call it sheaf cohomology.

If one combines both and allows the coefficient to be an “unstable sheaf” (an “$\infty$-stack”), one gets the notion of cohomology that we are using.

Posted by: Urs Schreiber on January 24, 2012 9:21 PM | Permalink | Reply to this

Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

Hi David,

somehow I had entirely missed this message of yours, and only saw Jim’s reaction to it.

What happens to the differential cohomological structures as you pass from one cohesive topos to another?

This is a good question, but I think it does not have and cannot have a general answer.

(But this is so for general reasons ;-), namely: going to the models of cohesion and then moving between them, if possible, precisely means to go from the abstract general to the concrete general, and then looking at specific coefficients for cohomology means to even go to concrete particular).

On the other hand, there are of course geometric morphisms between cohesive $\infty$-toposes along which it is very sensible to chase cocycles and cohomologies.

For instance Disc∞Grpd is refined by ETop∞Grpd is refined by Smooth∞Grpd, which in turn is refined both by SynthDiff∞Grpd as well as by SmoothSuper∞Grpd.

Undoing the second refinement means stripping off the “differential” aspect of all differential cohomology and retaining only continuous aspects, while undoing the first refinement means stripping off even the continuous aspects and just retaining homotopical information.

So there is a precise notion of “adding more refined cohesive structure”. This is important for the notion of “smooth refinement” of, say, characteristic maps: we identify a characteristic map $\mathbf{c}$ in $\mathrm{Smooth}\infty Grpd$ as a smooth refinement of an ordinary characteristic map $c$ in $Disc \infty Grpd$ by the fact that $\Pi(\mathbf{c}) \simeq c$.

Eventually, all the applications that I am motivated by have further refinements to $SuperSmooth\infty Grpd$. But we haven’t worked out very many details of that as of yet, unfortunately.

If we are allowed some outlandish speculation, is it possible that the commonality being found between geometric Langlands and number theoretic Langlands – e.g., use of Hitchin fibrations in the Fundamental Lemma, etc. – comes about because there are arithmetic forms of cohesiveness (such as your question ask for at MO, so that one should expect machinery that works in the case of smooth cohesion to carry over to arithmetic situations?

This is a great question, I think. That’s precisely the kind of speculation that I find is very natural in view of the available facts.

Unfortunately, at this point this is already all that I can say about this. Hopefully one fine day in the future we will see further.

Posted by: Urs Schreiber on February 2, 2012 10:59 PM | Permalink | Reply to this

Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

Perhaps a help for the latter question is David Nadler’s The geometric nature of the fundamental lemma. Parts of it are quite gentle, even for me.

Posted by: David Corfield on February 3, 2012 1:17 PM | Permalink | Reply to this

Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

I wonder if there might be other geometric morphisms than “adding more refined cohesive structure”, or will it turn out to be poset-like?

Posted by: David Corfield on February 3, 2012 1:28 PM | Permalink | Reply to this

Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

I wonder if there might be other geometric morphisms than “adding more refined cohesive structure”,

Yes, certainly one expects that.

or will it turn out to be poset-like?

No, this is just the present nascent state of the art.

I am strongly in application phase at the moment, looking at structures in the handful of models that I already have.

Later or elsewhere, somebody should spend more time on the space of all models of cohesion. (There is just not enough hours in the day!)

Posted by: Urs Schreiber on February 3, 2012 2:56 PM | Permalink | Reply to this

Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

I am somewhat amused when I think back on the flack I received when I first referred to `cohomological physics’.

Posted by: jim stasheff on January 24, 2012 2:13 PM | Permalink | Reply to this

Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

I am preparing some tables that are supposed to list twisting cocycles in nonabelian smooth cohomology together with their geometric interpretation in a way that helps getting an overview.