## December 23, 2017

### An M5-Brane Model

#### Posted by John Baez

When you try to quantize 10-dimensional supergravity theories, you are led to some theories involving strings. These are fairly well understood, because the worldsheet of a string is 2-dimensional, so string theories can be studied using 2-dimensional conformal quantum field theories, which are mathematically tractable.

When you try to quantize 11-dimensional supergravity, you are led to a theory involving 2-branes and 5-branes. People call it M-theory, because while it seems to have magical properties, our understanding of it is still murky — because it involves these higher-dimensional membranes. They have 3- and 6-dimensional worldsheets, respectively. So, precisely formulating M-theory seems to require understanding certain quantum field theories in 3 and 6 dimensions. These are bound to be tougher than 2d quantum field theories… tougher to make mathematically rigorous, for example… but even worse, until recently people didn’t know what either of these theories were!

In 2008, Aharony, Bergman, Jafferis and Maldacena figured out the 3-dimensional theory: it’s a supersymmetric Chern–Simons theory coupled to matter in a way that makes it no longer a topological quantum field theory, but still conformally invariant. It’s now called the ABJM theory. This discovery led to the ‘M2-brane mini-revolution’, as various puzzles about M-theory got solved.

The 6-dimensional theory has been much more elusive. It’s called the (0,2) theory. It should be a 6-dimensional conformal quantum field theory. But its curious properties got people thinking that it couldn’t arise from any Lagrangian — a serious roadblock, given how physicists normally like to study quantum field theories. But people have continued avidly seeking it, and not just for its role in a potential ‘theory of everything’. Witten and others have shown that if it existed, it would shed new light on Khovanov duality and geometric Langlands correspondence! The best introduction is here:

In a recent interview with Quanta magazine, Witten called this elusive 6-dimensional theory “the pinnacle”:

Q: I’ve heard about the mysterious (2,0) theory, a quantum field theory describing particles in six dimensions, which is dual to M-theory describing strings and gravity in seven-dimensional AdS space. Does this (2,0) theory play an important role in the web of dualities?

A: Yes, that’s the pinnacle. In terms of conventional quantum field theory without gravity, there is nothing quite like it above six dimensions. From the (2,0) theory’s existence and main properties, you can deduce an incredible amount about what happens in lower dimensions. An awful lot of important dualities in four and fewer dimensions follow from this six-dimensional theory and its properties. However, whereas what we know about quantum field theory is normally from quantizing a classical field theory, there’s no reasonable classical starting point of the (2,0) theory. The (2,0) theory has properties [such as combinations of symmetries] that sound impossible when you first hear about them. So you can ask why dualities exist, but you can also ask why is there a 6-D theory with such and such properties? This seems to me a more fundamental restatement.

Indeed, it sits atop a terrifying network of field theories in various lower dimensions:

Now, maybe, maybe this theory has been found:

If this holds water, it will be big.

Here’s the abstract:

Abstract. We present an action for a six-dimensional superconformal field theory containing a non-abelian tensor multiplet. All of the ingredients of this action have been available in the literature. We bring these pieces together by choosing the string Lie 2-algebra as a gauge structure, which we motivated in previous work. The kinematical data contains a connection on a categorified principal bundle, which is the appropriate mathematical description of the parallel transport of self-dual strings. Our action can be written down for each of the simply laced Dynkin diagrams, and each case reduces to a four-dimensional supersymmetric Yang-Mills theory with corresponding gauge Lie algebra. Our action also reduces nicely to an M2-brane model which is a deformation of the ABJM model.

My own interest in this is purely self-centered. I hope this theory holds water — I hope it continues to pass various tests it needs to pass to be the elusive (0,2) theory — because it uses ideas from higher gauge theory, and in particular the string Lie 2-algebra!

This is a ‘categorified Lie algebra’ that one can construct starting from any Lie algebra with an invariant inner product. It was first found (though not under this name) by Alissa Crans, who was then working on her thesis with me:

The idea was published here:

In 2005, together with Danny Stevenson and Urs Schreiber, we connected the string Lie 2-algebra to central extensions of loop groups and the ‘string group’:

• John C. Baez, Alissa S. Crans, Danny Stevenson, Urs Schreiber, From loop groups to 2-groups, Homotopy, Homology and Applications 9 (2007), 101–135.

though our paper was published only after a struggle. Subsequently Urs worked out a much better understanding of how Lie $n$-algebras appear in higher gauge theory and string theory. In 2012, together with Domenico Fiorenza, Hisham Sati, he began working out how string Lie 2-algebras are related to the 5-branes in M-theory:

They focused on the 7-dimensional Chern–Simons theory which should be connected to the elusive 6-dimensional (0,2) theory via the AdS/CFT correspondence. The new work by Saemann and Schmidt goes further by making an explicit proposal for the Lagrangian of the 6-dimensional theory.

For more details, read Urs’ blog article on G+.

All this leaves me feeling excited but also bemused. When I started thinking about 2-groups and Lie 2-algebras, I knew right away that they should be related to the parallel transport of 1-dimensional objects — that is, strings. But not being especially interested in string theory, I had no inkling of how, exactly, the Lie 2-algebras that Alissa Crans and I discovered might play a role in that subject. That began becoming clear in our paper with Danny and Urs. But then I left the subject, moving on to questions that better fit my real interests.

From tiny seeds grow great oaks. Should have stuck with the string Lie 2-algebra and helped it grow? Probably not. But sometimes I wish I had.

Posted at December 23, 2017 3:42 PM UTC

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### Re: An M5-Brane Model

What did you have to fight about to get that last paper published?

Posted by: Mike Shulman on December 23, 2017 9:30 PM | Permalink | Reply to this

### Re: An M5-Brane Model

Alissa, Danny, Urs and I first submitted our paper to a fancy-schmancy topology journal that engages in a multi-step refereeing process. The first referee was generally happy but thought the paper should be shortened. We did that, and then the second referee thought the paper had too much overlap with this one by André Henriques:

Abstract. Given an $n$-term $L_\infty$ algebra $L$, we construct a Kan simplicial manifold which we think of as the ‘Lie $n$-group’ integrating $L$. This extends work of Getzler math.AT/0404003. In the case of an ordinary Lie algebra, our construction gives the simplicial classifying space of the corresponding simply connect Lie group. In the case of the string Lie 2-algebra of Baez and Crans, this recovers the model of the string group introduced in math.QA/0504123.

That last reference is to the paper by Alissa, Danny, Urs and me! You see, Henriques gave talks about his work before we wrote our paper, but wrote this article afterwards, and had the kindness to cite us. Our approaches to getting ahold of the string group from the string Lie 2-algebra were technically quite different, so there was no real priority conflict. But the referee of that fancy-schmancy journal rejected our paper. And the whole process took a couple of years.

Posted by: John Baez on December 24, 2017 1:52 AM | Permalink | Reply to this

### Re: An M5-Brane Model

Mike wrote:

What did you have to fight about to get that last paper published?

I’ve added some more material to my article since you read it, including a cute chart… and I changed the wording from ‘fight’ to the more appropriate ‘struggle’.

Posted by: John Baez on December 24, 2017 2:35 AM | Permalink | Reply to this

### Re: An M5-Brane Model

I only had a superficial look at the Saemann-Schmidt paper, but I think there is a fundamental reason why any attempt at finding a classical action for the (2,0) is flawed. There is simply no parameter in the (2,0) theory that you can tune in order to obtain a semi-classical limit. (They actually mention this fact in the introduction, although I imagine they don’t consider as fatal as I do.)

In an ordinary gauge theory, you have a gauge coupling, and the weakly coupled region corresponds to the semi-classical limit. In the (2,0) theory, due to the self-duality of the higher gauge fields, the gauge coupling is fixed and the theory is always strongly coupled, or equivalently in the deep quantum regime.

I would guess that this is what Witten means when he says that “there is no reasonable classical starting point of the (2,0) theory”. Note that he didn’t say, “we don’t know a reasonable classical starting point of the (2,0) theory”. I think he means what he said.

There is an imperfect analogy in 2d. The (2,0) theory is similar to the level 1 WZW model. The difference is that in 6d there is no theory at higher level. Now we know that an action for the WZW model exists, but it is useful to describe the semi-classical limit, in which the level goes to infinity. It is completely useless to tackle the level 1 theory.

Posted by: Samuel Monnier on December 23, 2017 11:49 PM | Permalink | Reply to this

### Re: An M5-Brane Model

Saemann and Schmidt must have some thoughts about this objection; it would be interesting to know what they are.

Posted by: John Baez on December 24, 2017 2:30 AM | Permalink | Reply to this

### Re: An M5-Brane Model

In section 2 they give their reply to this, which is that the same problem can be raised for M2-brane models. But there we know a successful lagrangian model, ABJM, which has a discrete coupling in the CS level. A similar thing could happen for M5-branes.

Posted by: Jacob Winding on December 24, 2017 6:39 PM | Permalink | Reply to this

### Re: An M5-Brane Model

I wouldn’t buy that analogy without further explanations. The ABJM model describe M2-branes at the tip of a Z_k orbifold singularity. This is what makes possible a semi-classical limit, $k \rightarrow \infty$, where the action principle is meaningful.

See https://arxiv.org/abs/0806.1218 .

Posted by: Samuel Monnier on December 24, 2017 7:04 PM | Permalink | Reply to this

### Re: An M5-Brane Model

Our point is simply that a no-go-theorem that is known to fail for classical M2-brane models should not be seen as a big obstacle for classical M5-brane models. Note that our model contains a CS-like term (CH+BF^2), which requires us to restrict ourselves to discrete coupling constants. Also, the discreteness of our coupling constant is what circumvents the no-go-theorem of arXiv:hep-th/0004049 and arXiv:hep-th/9909094. So I think we agree that if a classical M5-brane model exists, it should have a discrete coupling constant. Our model does have this feature. The geometric interpretation of that discrete parameter should certainly be worked out and we have not done this yet.

Posted by: Christian Saemann on January 4, 2018 4:34 PM | Permalink | Reply to this

### Re: An M5-Brane Model

This makes me think looking at ADE orbifolds might be fruitful in this case…

Posted by: David Roberts on December 24, 2017 7:23 PM | Permalink | Reply to this

### Re: An M5-Brane Model

Is it known precisely how the discreteness of the allowed values for the coupling constant (or ‘level’) in Chern–Simons theory gets around this sort of argument, and also the simple scaling argument described by Jacques Distler below?

Of course the discreteness screws up any argument that assumes the coupling constant can take arbitrary values, but one might worry these arguments still apply in some sort of approximate way, especially in the classical limit (large levels).

Posted by: John Baez on December 24, 2017 6:52 PM | Permalink | Reply to this

### Re: An M5-Brane Model

There’s another objection (related to this one). It’s well-known that, upon circle-compactification, the 6D (2,0) theory reduces to 5D super-Yang-Mills.

If the 6D theory did have a Lagrangian formulation, you could do the standard Kaluza-Klein procedure: Fourier-expand all of the fields in $y$ (the coordinate on $S^1$), do the integral over $y$ (perhaps truncating to the low-lying Fourier-modes), and obtain a 5D action. If that action contained 5D Yang-Mills, you might be pretty happy for a minute or two.

But then you’d realize that you got the wrong answer. If the 5D Yang-Mills Lagrangian were obtained by integrating some 6D Lagrangian over the circle then you’d expect that the coupling constant in front, $\frac{1}{g_{\text{YM}}^2}$, would be proportional to the circumferences of the circle, $2\pi R$, just because that term was obtained by integrating something y-independent over the circle (that’s the only term in the Fourier series which survives the integral).

But the correct dependence of the 5D Yang-Mills coupling is $\frac{1}{g_{\text{YM}}^2}\propto \frac{1}{R}$. That’s a very puzzling result to come out of any dimensional-reduction of a Lagrangian field theory.

(S&S talk about the reduction of their theory from 6 to 4 dimensions. I don’t understand their discussion, but presumably there’s a reason they skipped over the conceptually-simpler case of the reduction to 5D.)

Posted by: Jacques Distler on December 24, 2017 5:08 AM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

Actually, their $T^2$ compactification down to 4D has pretty much the same problem.

Assume, for simplicity, that the $T^2$ is obtained by identifying sides of a rectangle – that is, that $\tau$ is pure imaginary. (If it’s obtained by identifying sides of a parallelogram, then the Fourier-analysis is a little more complicated, but one can still carry through the same analysis, with the same conclusion.)

Fourier-decompose all the fields with respect to $y_1,y_2$. First, we reduce with respect to $y_2$, to obtain 5D super-Yang-Mills, with gauge coupling $\frac{1}{g_5^2}\propto \frac{1}{R_2}$. Now that we have an ordinary Lagrangian field theory, the second step of reducing with respect to $y_1$ proceeds in the ordinary Kaluza-Klein fashion with the result that the 4D SYM coupling, $\frac{1}{g_4^2}\propto \frac{R_1}{g_5^2}\propto \frac{R_1}{R_2}$.

Of course, we could have reduced on the two circles in the opposite order, obtaining a super-Yang-Mills theory with coupling $\frac{1}{\tilde{g}_4^{2}}\propto \frac{R_2}{R_1}$. That’s the basic statement of S-duality.

So how do S&S obtain a 4D gauge coupling $\frac{1}{g_4^2}\propto \frac{R_1}{R_2}$ from the Kaluza-Klein reduction of a Lagrangian field theory in 6D? We know that each term in the Lagrangian has a reduction proportional to $R_1 R_2$. The term they’re interested in has a scalar field multiplying the Yang-Mills term: $\mathcal{L} \sim \dots + \phi |F|^2+\dots$. They then just assert that $\phi$ has a VEV $\propto \frac{1}{R_2^2}$. This assertion has a host of problems.

• If $\phi$ has a flat direction, then any value for its VEV is as good as any other. There’s no rationale for this particular fine-tuned nonzero value.
• This prescription clearly breaks the S-duality symmetry which, in our simple setup, was just the geometrical symmetry $y_1\leftrightarrow y_2$.
• It also breaks conformal invariance, and it’s unclear what fine-tuning recovers the conformal invariance of the 4D theory.
Posted by: Jacques Distler on December 24, 2017 8:57 AM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

We are certainly aware of this argument, but we think we can circumvent it, at least the scaling part. Giving phi an expectation value is not as crazy as it sounds: This is exactly the mechanism used in reducing M2-branes to D2-branes, see arXiv:0803.3218. There, the scalar field also has a flat direction. What is strange in our case is that phi_s acquires this expectation value. There is certainly more work do be done to motivate this better from a string theoretic perspective. However, we show at least that this scaling argument can be circumvented and needs to be complemented by a new argument to still exclude a reduction to super Yang-Mills theory. (This is the minimum result of our paper: many arguments supporting no-go-theorems freely used in the literature are actually more subtle and need refinement to hold; at the moment, I believe that they will collapse.)

Now to S-duality: The reason we restrict to 4d is clearly that we need phi to be a coupling constant after integration over the reducing torus, so we need to end up in 4d. Another reason is, retrospectively, that we end up on a D3-brane which is self-dual under S-duality. This is important since the torus modulus is broken up into real part and imaginary part, and both become vevs or backgrounds for two distinct fields in our model. Note that we do get N=4 SYM (I think the coupling of the two N=2 multiplets is the right one) in 4d with theta-term F^2. Would you expect anything more?

Why is conformal invariance broken? Let’s say we do have N=4 SYM there (again, I belief the coupling between the two N=2 multiplets to be the right one), then the theory is conformal.

Posted by: Christian Saemann on January 4, 2018 4:53 PM | Permalink | Reply to this

### Re: An M5-Brane Model

Jacques wrote:

But the correct dependence of the 5D Yang-Mills coupling is $\frac{1}{g_{\text{YM}}^2}\propto \frac{1}{R}$.

How does the correct calculation go, roughly?

Since assuming the theory comes from a Lagrangian gives $\frac{1}{g_{\text{YM}}^2}\propto R$ instead of $\frac{1}{g_{\text{YM}}^2}\propto \frac{1}{R}$, I’m thinking about the transformation

$R \mapsto \frac{1}{R}.$

Could the 6d theory on a Riemannian 5-manifold $M$ times a circle of radius $R$ be isomorphic to some other 6d theory described by a Lagrangian on $M$ times a circle of radius $1/R$?

Just messin’ around here…

Posted by: John Baez on December 24, 2017 5:26 PM | Permalink | Reply to this

### Re: An M5-Brane Model

How does the correct calculation go, roughly?

There are many ways to see this, but the (drop-dead) simplest argument is just dimensional analysis: the 6D theory is conformally-invariant. It has no dimensionful parameters. The 5D theory has a characteristic mass-scale, $\frac{1}{g^2}$. Where does that come from? Well, there’s only one dimensionful scale in the problem, namely $R$. Hence, by trivial dimensional analysis, $\frac{1}{g^2}\propto \frac{1}{R}$.

A more detailed argument involves studying the BPS particles (“$W$-bosons”) of the 5D theory out on the Coulomb branch.

Both the 5D SYM theory and the 6D (2,0) theory have a moduli space of vacua of dimension $5r$, where $r$ is the rank of the ADE Lie algebra. With some abuse of nomenclature, this is usually called the Coulomb branch. At a generic point on the Coulomb branch the 5D gauge symmetry is broken to the Cartan. The gauge bosons corresponding to the broken generators are massive and their masses are BPS-protected, so their tree-level dependence on the Yang-Mills coupling is exact.

On the other hand, these BPS particles have a 6D origin as BPS strings, wrapped on the circle. When you compare the two BPS mass formulæ, you can read off the precise formula for the Yang-Mills coupling as a function of $R$.

I’m thinking about the transformation $R\mapsto\frac{1}{R}.$

6D Little String Theory has T-duality. The 6D (2,0) theory does not. Indeed, that was the whole point which lead Seiberg and others in the 1990s to conclude that it’s a local field theory (with a unique local stress-energy tensor) and not a string theory.

Posted by: Jacques Distler on December 24, 2017 7:45 PM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

Yet another test is described on slide 35 of this talk by Tachikawa: http://physics.princeton.edu/strings2014/slides/Tachikawa.pdf

Essentially, if you reduce the 6d (2,0) SU(2N) theory on a circle with a Z_2 twist, you should get a SO(2N+1) 5d SYM. This would be hard to achieve in a semi-classical setting, because SO(2N+1) is not a subgroup of SU(2N).

Posted by: Samuel Monnier on December 24, 2017 6:52 PM | Permalink | Reply to this

### Re: An M5-Brane Model

Not even their 6D description makes any sense.

The 6D (2,0) theory is supposed to have $5r$ dimensional moduli space of vacua. At a generic point on that moduli space, the low-energy physics is that of $r$ free (2,0) tensor multiplets. The lattice of BPS string charges is the $r$-dimensional ADE root lattice, and the BPS strings correspond to the positive roots.

Their theory has only a single tensor multiplet (the self-dual 3-form field strength is denoted $\mathcal{H}$ in their paper). So, while there are lots of other fields (whose role is … ahem … unclear), they don’t seem to have the tensor multiplets required to parametrize the Coulomb branch of the theory (i.e., the only part of the 6D physics that we can honestly say that we completely understand).

At this point, I am completely mystified by Urs’ enthusiasm.

Posted by: Jacques Distler on December 24, 2017 10:17 PM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

“If this holds water, it will be big” doesn’t sound like enthusiasm to me. It sounds like caution.

I’m curious how many of the objections to the new Saemann–Schmidt paper are also applicable to this earlier one:

Posted by: John Baez on December 24, 2017 10:32 PM | Permalink | Reply to this

### Re: An M5-Brane Model

I believe the paper of Fiorenza-Sati-Schreiber describes (using the string Lie algebra) a 7d TFT which is expected to admit the (2,0) theory as a boundary condition – or in the language of Freed-Teleman, who also study the same general story, the (2,0) theory is expected to be defined not as an absolute field theory but as a theory relative to this 7d Chern-Simons theory. It doesn’t propose a precise model for the (2,0) theory itself.

Posted by: David Ben-Zvi on December 24, 2017 11:54 PM | Permalink | Reply to this

### Re: An M5-Brane Model

This is certainly in the top five of the problems of our model that we plan to address as soon as possible. Clearly, n well-separated M5-branes should be described by n abelian tensor multiplets. In principle, we could take n copies of the string Lie 2-algebra of u(n) and plug this in our model. This has the exciting property that the degrees of freedom scale as n^3 (from the n copies of u(n)). Annoyingly, the n copies wouldn’t talk to each other, and we haven’t found a way yet of making this happen. (We are certainly thinking about this!)

When writing up our model, we decided to limit ourselves to a gauge structure that reproduces the full SYM theory in 4d and the full Chern-Simons matter theory in 3d. Note also that since the M-theory Coulomb branch should reduce to the SYM Coulomb branch in 4d (which we have in our model with our gauge structure: just take string of u(1)^n as gauge structure), there must be something more interesting happening.

Posted by: Christian Saemann on January 4, 2018 10:58 PM | Permalink | Reply to this

### Re: An M5-Brane Model

I’m a little confused here. On the one hand, you want to claim that (after compactifying to 4D), your model (as written) reproduce the full nonabelian nature of the 4D $\mathcal{N}=4$ SYM.

On the other hand, your model (as written) doesn’t have the correct degrees of freedom to parametrize the Coulomb (or, more properly “tensor”) branch of the 6D (2,0) theory (whose low-energy physics is that of $r$ abelian tensor multiplets).

Since the tensor branch of the 6D theory dimensionally-reduces to (a $5r$-dimensional subspace of the $6r$-dimensional) Coulomb branch of the 4D theory, this seems like a contradiction.

To make this more precise, the gauge bosons of the Cartan subgroup (which is unbroken at a generic point on the Coulomb branch) are supposed to be the dimensional reduction of the $r$ tensor multiplets (that don’t appear in your model); the massive “W-bosons”, corresponding to the broken generators of the gauge group, are wrapped BPS strings of the 6D theory (because you don’t have the tensor multiplets, those strings are either absent or not BPS, because they’re not charged under the relevant (but in your model absent) 1-form symmetries).

If, somehow, you managed to add the degrees of freedom necessary to get the 6D physics right, you would be in danger of getting too many Yang-Mills-like degrees of freedom upon compactification to 4D (the ones you already claim to have, plus the new ones arising from the new degrees of freedom you’ve added).

Before getting to the (rather more subtle) 4D physics, you need to get the 6D and 5D physics right. Your model doesn’t seem to have the right degrees of freedom to do that.

Finally, as I pointed out, there is a rich class of 6D theories which (at least, superficially) look a lot more like your model, namely (1,0) SCFTs. But the restrictions (from anomalies and other considerations) on those theories are rather severe (which, ultimately, is what allowed Heckman et al to classify them).

Have you, for instance, checked the most basic condition, namely the absence of 6D gauge anomalies in your model?

Posted by: Jacques Distler on January 9, 2018 7:47 PM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

I wrote

…your model (as written) doesn’t have the correct degrees of freedom…

Let me spell that out a little bit more.

We’ve already discussed the tensor multiplets (or shortage thereof). But your model does have a scalar field, $\phi$, multiplying $tr(F^2)$. That’s the scalar field to which you give a fine-tuned VEV, when you compactify to 4 dimensions.

Whatever …

It has a flat potential, so we can give it any VEV we want. Let’s do that … in 6 dimensions. When $\langle\phi\rangle$ is large, the effective Yang-Mills coupling is small. So, out on this branch of the moduli space of vacua of your model, we have weakly coupled gauge bosons (and other stuff).

But there’s no place on the moduli space of vacua of the (2,0) theory where there are weakly-coupled gauge bosons. There can’t be; that would be incompatible with (2,0) supersymmetry.

The only massless multiplet of (2,0) supersymmetry is the tensor multiplet. You don’t have enough of those; instead, you do have (weakly-coupled, on a branch of the moduli space of vacua) a (1,0) Yang-Mills multiplet – something which is incompatible with (2,0) supersymmetry.

Posted by: Jacques Distler on January 10, 2018 3:32 AM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

Thanks for your comments, Jacques, they are very much appreciated. I fully agree that we do not seem to have the right degrees of freedom for the Coulomb branch. And indeed, if we naively added them, we would get too many, as we already have the right degrees of the Coulomb branch of the 4d theory. This is the problem we’re working on at the moment.

However, I do think we have the right degrees for a pair of M5-branes, and that the slogan “string(3) is the categorified analogue of su(2)” seems to be correct: we have two tensor multiplets labeled r and s and they should correspond to center-of-mass BU(1) and relative BU(1) when moving the pair of M5-branes apart.

Again, we are already rather excited that we found a theory that checks quite a few boxes for a classical (2,0) theory: Mathematically well-defined notions of interacting B-fields (as opposed to the usual ad-hoc constructions), SUSY action with the right field content (at least for 2 M5-branes), a straightforward reduction mechanism to super Yang-Mills, even if “only” in 4d and a straightforward reduction to Chern-Simons matter. The problems we list towards the end of our introduction suggest that this our model is not the last word and more work needs to be done.

As far as anomalies are concerned, our model is suffering from the usual “gauge anomaly” of the action functional (as Chern-Simons theory), because it contains a higher Chern-Simons term. Just as for Chern-Simons theory, this restricts the total coupling constant to discrete values. For true quantum anomalies, we haven’t performed any further checks; our considerations are purely classical, nobody successfully quantized an interacting non-abelian higher gauge theory yet (another project that is in our pipeline).

I don’t get your concern about the flat directions: Isn’t literally the same happening in the M2 to D2 reduction of arXiv:0803.3218? Sure, the gauge fields have a potential, but the choice of VEV in their case is also free up to a scale. So our vev should be justified in more detail from a string theory perspective, but it does not seem surprising.

I do not fully understand the appropriate interpretation of the scalar fields in the tensor multiplet, but it seems to me that the full (2,0)-theory should exist only at $\phi=0$, which our model can’t capture for various reasons (as we write, the PST formalism collapses there, and even the equations of motions become essentially free). In arXiv:1212.5199, at the bottom of page 3, they argue that this behavior is due to the tensionless string phase transition, but I do not know enough to comment on this much further. I clearly have to study this in more detail but currently, I think our model has a chance of being a “(2,0)-theory with slightly broken symmetry”.

Posted by: Christian Saemann on January 10, 2018 11:58 AM | Permalink | Reply to this

### Re: An M5-Brane Model

However, I do think we have the right degrees for a pair of M5-branes …

I don’t see that. For 2 M5-branes, you (at least) do have two tensor multiplets. But:

• The metric for their kinetic terms is Lorentzian (signature (1,1)), rather than Euclidean.
• In the correct description, one of the tensor multiplets should should be decoupled (representing the center-of-mass motion of the two M5-branes.

On the other hand, you also have a Yang-Mills multiplet, which should not be there (for reasons already discussed).

So, no, I don’t think this is a correct description, even of the $A_1$ (2,0) theory.

… a straightforward reduction mechanism to super Yang-Mills,

But not the one you want.

What you have is a theory with super Yang-Mills in 6 dimensions. Sure that will reduce to super Yang-Mills in 4. But that’s not how super Yang-Mills arises in the reduction of the (2,0) theory.

The fact that you have Yang-Mills in 6D should be telling you that you are on the wrong track, not the right one.

Both your model and the (2,0) theory have a moduli space of vacua in 6D. But they are manifestly different moduli spaces of vacua, with different infrared physics along them. While it’s hard to compare the theories directly at $\langle\phi\rangle=0$, the fact that the physics along the moduli spaces of vacua are completely different should be convincing evidence that the theories themselves are completely different.

Posted by: Jacques Distler on January 10, 2018 4:58 PM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

This is an interesting point and certainly a relevant consistency check. There may be some enhancement mechanism of gauge symmetry by choosing some particular gauge structure and compactifying, but we have not thought about this.

At the moment, we have quite a few other checks to perform, also related to higher parallel transport, which seem to us more basic. Once things work there, we’ll certainly return to this issue. Note that we are not certain yet about the best choice of gauge group underlying our string-like Lie 4-algebra, neither are we sure that we found the best gauge structure for hypermultiplets. There is quite some room for trial-and-error here. We merely suggested some that allow for the most obvious consistency checks to pass.

Posted by: Christian Saemann on January 4, 2018 5:01 PM | Permalink | Reply to this

### Re: An M5-Brane Model

Could someone explain how introducing the string Lie algebra, which if I understand correctly is a higher central extension of a simple Lie algebra, can help with the basic problem of understanding the (2,0) theory, which as I understand it has to do with delooping a compact Lie group? In the abelian case, which is relatively well understood, the (2,0) theory is a BT gauge theory, i.e. related to maps to BBT. Out on the Coulomb branch of the (2,0) theory for G it likewise becomes a BT gauge theory. But at the conformal point it would seem to be a theory of “BG bundles”, or maps to a nonexistent space BBG. The string Lie algebra is clearly an important and fascinating object which will surely play a role in the eventual answer, but at a cursory glance the emphasis on its role in understanding what kind of a beast is the (2,0) theory seems to me misplaced.

Posted by: David Ben-Zvi on December 24, 2017 10:12 PM | Permalink | Reply to this

### Re: An M5-Brane Model

David Ben-Zvi wrote:

Could someone explain how introducing the string Lie algebra, which if I understand correctly is a higher central extension of a simple Lie algebra, can help with the basic problem of understanding the (2,0) theory, which as I understand it has to do with delooping a compact Lie group?

I wish I could explain the hope here, which seems to motivate not only Saemann and Schmidt’s paper An M5-brane model but also Fiorenza, Sati and Schreiber’s paper Multiple M5-branes, string 2-connections, and 7d nonabelian Chern-Simons theory. But I don’t know enough to say anything interesting, so all I can do is ask questions and quote stuff.

Has anyone here looked at the latter paper? David wrote:

I believe the paper of Fiorenza–Sati–Schreiber describes (using the string Lie algebra) a 7d TFT which is expected to admit the (2,0) theory as a boundary condition – or in the language of Freed–Teleman, who also study the same general story, the (2,0) theory is expected to be defined not as an absolute field theory but as a theory relative to this 7d Chern-Simons theory. It doesn’t propose a precise model for the (2,0) theory itself.

I’ll just quote the start of that paper:

The quantum field theory (QFT) on the worldvolume of M5-branes is known [Wi04, HNS, He] to be a 6-dimensional (0, 2)-superconformal theory that contains a 2-form potential field $B_2$, whose 3-form field strength $H_3$ is self-dual (see [Mo] for a recent survey). 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 [Wi04], geometric Langlands duality [Wi09], and Khovanov homology [Wi11]. 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 (a gerbe is a “higher analog” of a gauge bundle, discussed in detail below in section 3). See [Ha, Be, Sa10b] for surveys of the problem and recent developments. Here we add another piece to the scenario, by proposing a 7d Chern-Simons theory which appears to be a natural candidate for the holographic dual of the multiple M5-branes 6d QFT via $AdS_7$/$CFT_6$-duality, and by identifying the nonabelian 2-form fields appearing in the theory as local data of (twisted) String 2-connections.

Namely, for a single M5-brane, the Lagrangian of the theory has been formulated in [HSe, PeS, Sch, PST, APPS] and in this case there is, due to [Wi96], a holographic dual description of the 6d theory by 7-dimensional abelian Chern-Simons theory, as part of $AdS_7$/$CFT_6$-duality (reviewed for instance in [AGMOO]). We give here an argument, following [Wi96, Wi98b] but taking the quantum anomaly cancellation of the M5-brane in 11-dimensional supergravity into account, that in the general case: the $AdS_7$/$CFT_6$-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, as considered in [SSS09a, FSS10]. 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 $\mathfrak{e}_8$. We also describe the global structure of the moduli 2-stack of field configurations, which is more subtle.

I don’t know anything about this stuff, so I could easily be completely wrong, but it sounds like they’re proposing a 7d theory as a natural candidate for the “holographic dual” of the “multiple M5-branes 6d QFT” via $AdS_7$/$CFT_6$-duality. Isn’t the “multiple M5-branes 6d QFT” the mysterious (2,0) theory?

Posted by: John Baez on December 28, 2017 9:21 PM | Permalink | Reply to this

### Re: An M5-Brane Model

A quick summary of why the string Lie 2-algebra is interesting and relevant; more enlightening details in arXiv:1201.5277, 1705.02353 and, hopefully, 1712.06623.

1) If there is a classical (2,0)-theory, it is a higher gauge theory. These have categorified Lie algebras or Lie 2-algebras as gauge algebras, and the string Lie 2-algebra is one of them. Note that the string Lie 2-algebra can be regarded as a non-abelian extension of BU(1).

2) The string Lie 2-algebra exists for each ADE-type Lie algebra, which is good news for describing (2,0)-like theories.

3) There are actually not many examples of interesting Lie 2-algebras. This is due to the large equivalence classes of L_infty-algebras. John Baez and Alissa Crans have shown in arXiv:math.QA/0307263 that any Lie 2-algebra is of the form $V \mapsto \mathfrak{g}$, for $V$ a vector space and $\mathfrak{g}$ a Lie algebra. A triple product is then encoded in $H^3(\mathfrak{g},V)$. The string Lie 2-algebra is the simplest such example and, except for infinite dimensional cases, I’m not aware of any other interesting ones.

4) In physics, the first interesting non-abelian gauge group to study is Spin(3)=SU(2). String(3) is the categorified analogue of this, which one can argue from many perspectives. One of my favourite ones (since it finds application in monopoles and higher monopoles) is that SU(2), as a manifold, is the total space of the Hopf bundle over S^2, the fundamental principal bundle over S^2 with unit first Chern class. String(3), as a 2-space, is the total categorified space of the fundamental abelian gerbe over S^3 with Dixmier-Douady class 1.

5) Note that there is a categorically equivalent loop space model of the string group, which links to many results from physics (e.g. need for central extension of \Omega E_8 as higher gauge structure to cancel anomalies, etc.).

6) Perhaps a little special, but again related to things I’m interested in: In the fuzzy funnel developing when D1-branes end on D3-branes, quantized 2-spheres emerge and the underlying Hilbert space has an action of Spin(3) (the double cover of the isometries on the sphere) on it. We argued in arXiv:1608.08455 that the string group acts on the categorified Hilbert space arising in the quantization of 3-spheres, which are expected to arise when M2-branes end on M5-branes.

Posted by: Christian Saemann on January 4, 2018 10:14 PM | Permalink | Reply to this

### Re: An M5-Brane Model

Thanks for joining the conversation, Christian!

The string Lie 2-algebra exists for each ADE-type Lie algebra, which is good news for describing (2,0)-like theories.

Just a technical note: for any simple real Lie algebra $\mathfrak{g}$ we have

$H^3(\mathfrak{g},\mathbb{R}) = \mathbb{R}$

where the first $\mathbb{R}$ here stands for the trivial 1-dimensional representation of $\mathfrak{g}$. So, the classification result you mention gives a 1-parameter family of string Lie 2-algebras for each such $\mathfrak{g}$. This parameter is the ‘level’, which also parametrizes central extensions of loop groups (or their Lie algebras).

I would love to know something special about the string Lie 2-algebras of ADE Lie algebras. (I know tons of other nice things about these Lie algebras. Presumably to answer my question we’d have to get the root lattice involved.)

The string Lie 2-algebra is the simplest such example and, except for infinite dimensional cases, I’m not aware of any other interesting ones.

A bunch of others are discussed here:

One reason there are more is that one can get a Lie 2-algebra either from an infinitesimal crossed module or a 3-cocycle on a Lie algebra $\mathfrak{g}$ taking values in any representation of $\mathfrak{g}$, and Lie 2-algebras that are finite-dimensional in one picture are often not finite-dimensional in the other picture.

Posted by: John Baez on January 4, 2018 10:46 PM | Permalink | Reply to this

### Re: An M5-Brane Model

Yes, thanks for reminding me. Of course there are other examples, but they are harder to sell to physicists and they do not seem “interesting” to me (I may well be completely wrong here).

For example, the fact that any representation of a Lie algebra leads to a Lie 2-algebra is nice, but the only interesting use for it that I found is that we can use it to turn the M2-brane model into a higher gauge theory with some brute force introduction of Lagrange multipliers (arXiv:1311.1997). But then again, your application of the Poincare Lie 2-algebra to things like teleparallel gravity sounds very interesting indeed.

But for the moment: Do you happen to know an example of a skeletal Lie 2-algebra with interesting 3-bracket, which is finite dimensional and not the string Lie 2-algebra or a straightforward variant thereof?

And yes, absolutely, from the perspective of the string Lie 2-algebra, there’s nothing special about ADE. We just have enough room for what we might want. Unfortunately, I do not see any way of getting the restriction to ADE from classical considerations, but then again I should stress that I haven’t tried very hard.

Posted by: Christian Saemann on January 4, 2018 11:19 PM | Permalink | Reply to this

### Re: An M5-Brane Model

Christian wrote:

Of course there are other examples, but they are harder to sell to physicists and they do not seem “interesting” to me (I may well be completely wrong here).

Different people have different concepts of “interesting”, so I’ll just list the examples from my paper with John Huerta and let everyone decide for themselves if they’re interested:

First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2-group’, which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2-group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2-group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2-group’.

After that paper, it became clear that the Poincaré 2-group also shows up in a formulation of general relativity first studied by Cartan and Einstein:

Do you happen to know an example of a skeletal Lie 2-algebra with interesting 3-bracket, which is finite dimensional and not the string Lie 2-algebra or a straightforward variant thereof?

I don’t know any. For the mathematicians in the room, I’ll point out that this is equivalent to the following:

Puzzle. Is there a finite-dimensional Lie algebra $\mathfrak{g}$ and a finite-dimensional representation $V$ for which $H^3(\mathfrak{g},V) \ne 0$, but which doesn’t arise from the case where $V$ is the trivial representation via some straightforward tricks?

To answer this, I’d have to start by figuring out all the “straightforward tricks”.

Posted by: John Baez on January 5, 2018 12:48 AM | Permalink | Reply to this

### Re: An M5-Brane Model

Thanks for the reformulation, this is indeed what I would be interested in. What I would count as straightforward tricks are things like taking the obvious direct sums or trivially completing to cotangent space (as necessary for having a symplectic form which encodes a metric).

Particularly interesting are Lie 2-algebras $V\rightarrow \mathfrak{g}$, where we have a symmetric map $(-,-): \mathfrak{g}\times\mathfrak{g} \rightarrow V$ and the 3-bracket $\mu_3$ of the Lie 2-algebra given by $(-,[-,-])$. These are essentially the ones that can be extended and plugged into the model of Samtleben et al., which underlies our constructions.

Posted by: Christian Saemann on January 5, 2018 9:44 PM | Permalink | Reply to this

### Re: An M5-Brane Model

There’s a paper of Friedrich Wagemann on crossed modules of Lie algebras that gives some broad no-go theorems that are useful for ruling out examples.

Posted by: David Roberts on January 5, 2018 11:28 PM | Permalink | Reply to this

### Re: An M5-Brane Model

There are actually not many examples of interesting Lie 2-algebras.

Does the T-Duality Lie 2-algebra count as interesting (section 7 of this)?

Since the T-duality 2-group is

the string 2-group for the cup product universal characteristic class on fiber products of torus-fiber bundles with their dual torus bundles,

perhaps this reinforces the thought that there’s not much choice.

Posted by: David Corfield on January 5, 2018 3:05 PM | Permalink | Reply to this

### Re: An M5-Brane Model

The T-Duality Lie 2-algebra seems “too abelian” to be interesting for interacting M5-branes.

There is, however, a link to T-duality: The local symmetries of supergravity coupled to a 2-form gauge potential (i.e. the massless sector of string theory) are given by diffeomorphisms and gauge transformations of the Kalb-Ramond field $B$, which is part of the connection of an abelian gerbe. These are roughly parameterized by sections of the exact Courant algebroid $TM\oplus T^*M$. The Courant algebroid is a symplectic Lie 2-algebroid, as Roytenberg and Urs keep stressing. It is also the correct setting for describing T-duality. Symplectic Lie n-algebroids come with an associated Lie n-algebra (a very beautiful construction via derived brackets). For the exact Courant algebroid, this Lie 2-algebra is a form of semidirect product of the diffeomorphisms and the gauge transformations. That is, the Lie 2-algebra is the Lie 2-algebra of symmetries of the supergravity (details are, e.g., in arXiv:1611.02772).

The Lie 2-algebra associated to the Courant algebroid of $S^3$ has the string Lie 2-algebra $\mathfrak{string}(3)$ as a Lie 2-subalgebra as shown by John and Chris Rogers (arXiv:0901.4721).

If one starts from heterotic supergravity (e.g. arXiv:1711.03308), one even gets the semidirect product of a string Lie 2-algebra and the diffeomorphisms.

By now, I do have the feeling that the set of interesting/relevant Lie 2-algebras is indeed very limited. At least, the same ones seem to crop up everywhere.

Posted by: Christian Saemann on January 5, 2018 10:02 PM | Permalink | Reply to this

### Re: An M5-Brane Model

I think the idea in this paper is not that they have found a Lagrangian for (2,0) theory, but that they have found a Lagrangian for a field theory that has some significant similarities. The theory is described as an M5-brane “model”, and the abstract simply states “We present an action for a six-dimensional superconformal field theory containing a non-abelian tensor multiplet”, not “We have found a Lagrangian description of (2,0) theory”.

Posted by: Mitchell Porter on December 26, 2017 10:41 PM | Permalink | Reply to this

### Re: An M5-Brane Model

First of all, I would not call what they have written down a “non-abelian” tensor multiplet. It’s pair of abelian tensor multiplets with an indefinite-signature kinetic term (already a sign of a nonunitary theory), coupled to various other fields.

Secondly, 6D superconformal field theories with at least (1,0) superconformal invariance were classified by Heckman, Morrison, Rudelius and Vafa. Some of them have “Lagrangian” formulations superficially similar to (but not the same as) this one. The basic example (the only one which is a purely “Lagrangian”) is a linear quiver gauge theory where each node is an $SU(N)$ factor in the gauge group, each edge is a bifundamental hypermultiplet, and there’s a tensor multiplet coupled to each gauge node.

$N$ I put “Lagrangian” in quotes, because it is not a useful formulation of the theory at the conformal point (which is located at $\phi_i=0$, where the Lagrangian becomes singular). Rather, the Lagrangian is useful out on the Coulomb branch (where $|\langle\phi_i\rangle|\gg 0$).

Of course, Heckman et al assumed unitarity. If you don’t demand unitarity (and this theory certainly is not unitary), then presumably there are many more solutions, some of which are Lagrangian.

Thirdly, the body of the paper not only asserts to be a formulation of the (2,0) theory (as opposed to some random non-unitary (1,0) SCFT), but claims to have performed many checks of that assertion.

None of those claims seem to hold water…

Posted by: Jacques Distler on December 27, 2017 6:16 AM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

The canonical nomenclature (as far as I’m aware of) is to call the usual gerbes in any of the essentially equivalent forms as defined by Gawedzki, Murray, Hitchin, etc. “abelian”. This is because they are the direct categorified analogues of principal U(1)-bundles. Consequently, we should call all other categorified principal bundles “non-abelian”. The B-field in the tensor multiplet is therefore part of the connective structure of a non-abelian gerbe or principal 2-bundle with structure 2-group the string 2-group. One can now argue whether this warrants the name “non-abelian tensor multiplet”, but this is hairsplitting. At the end of the day, abelian vs non-abelian essentially stands for free vs interacting in the context of the (2,0)-theory. Since our theory includes non-abelian SYM theory in a particular reduction (whether or not this reduction is string theoretically sensible is another issue) it is clearly an interacting theory.

I agree that the first impression suggests that the string Lie 2-algebra is “not really” non-abelian, and I believe that quite a few people reached the same conclusion. However, they fail to notice that it is, in fact, sufficient, e.g. for getting an interacting theory containing a tensor multiplet that contains N=4 SYM in a limit. By now, I’m relatively certain that if there’s a classical structure that should underly the (2,0)-theory, it’s the string Lie 2-algebra (or the Lie 4-algebra extension of it we used in the paper). In particular, the existence of higher and interacting monopoles obatined in our previous paper arXiv:1705.02353 convinced me.

Note that principal 2-bundles with the string 2-group as structure 2-group (i.e. “string structures”) are not unusual at all: They underly heterotic string theory as well as the global description of stacks of D-branes in the presences of a topologically non-trivial B-field and lead to twisted K-theory.

The issue of the indefinite-signature kinetic term is more serious. We do not have a good answer yet, but we hope to come up with a similar solution as in the case of M2-brane models based on Lorentzian 3-Lie algebras. The other issue we mention in the paper, which is equally problematic, is that the PST formalism in the action only works for phi!=0. Otherwise, one only has equations of motion. Again, we do not have a good answer yet.

Your last paragraph seems quite exaggerated and rather unfair. If you should have any particular passage that comes to mind, please let us know; we’d be happy to tone it done. Note that if we had been thinking we had the (2,0)-theory, we would have stated so in the abstract.

Again: We have some model that passes quite a few consistency checks and shows that the handwaving no-go-theorems in the literature have plenty of holes. I believe we do say rather clearly, that our model may not be a relevant model (see first paragraph of the introduction); in particular, we point out important shortcomings we identify (see paragraph starting “There are a number of remaining open problems concerning our M5-brane model…” on page 4).

Of course, we do believe that our model is a relevant contribution towards finding a classical (2,0)-theory. If one doesn’t think so, one might easily misread our paper.

Posted by: Christian Saemann on January 4, 2018 10:46 PM | Permalink | Reply to this

### Re: An M5-Brane Model

This is indeed our point of view. Thanks for confirming that our preprint can be understood this way.

Posted by: Christian Saemann on January 4, 2018 10:18 PM | Permalink | Reply to this

### Re: An M5-Brane Model

On a somewhat digressive note, I’m wondering what examples we have at all of quantum field theories in any dimension that don’t arise from quantizing classical field theories defined by Lagrangians?

The constructive quantum field theorists are completely open to this possibility, and Stephen Summers writes:

To conclude, in my view the heroic efforts of constructive quantum field theorists in the 70’s and 80’s took semiclassical ideas to their technical limits. Since they did not succeed in fully constructing interacting quantum field models in four spacetime dimensions, it would appear that a different approach to constructing models must be found. Although workers in the field are far from reaching the holy grail of constructing, say, the Standard Model (if it even exists), what is heartening about the results outlined above is the fact that most of these new models cannot be constructed using the older ideas, either because the technical effort to do so is prohibitive or because it is quite simply excluded mathematically. Indeed, most of these models do not seem to have an associated Lagrangian. Therefore, one is truly treading upon new ground.

Posted by: John Baez on December 28, 2017 9:37 PM | Permalink | Reply to this

### Re: An M5-Brane Model

Over on G+, Urs Schreiber and I dreamt up an absurdly simple example of a QFT that comes from quantizing a linear classical field theory that does not arise from a Lagrangian.

I’d call this classical field theory the ‘left-moving wave equation’. Our field is a function

$\phi : \mathbb{R}^2 \to \mathbb{R}$

where $\mathbb{R}^2 \ni (t,x)$ is 2d Minkowski spacetime. The field equation is

$\partial_t \phi = \partial_x \phi$

The solutions are just functions of the form

$\phi(t,x) = f(t+x).$

This field theory is Lorentz-invariant, and it’s easy to quantize: in fact, I often use it when teaching kids quantum field theory!

But it doesn’t come from a Lagrangian. Urs explained:

The usual Lagrangian $|\nabla \phi|^2$ gives two independent copies of this system, one left moving, the other right moving, and discarding either copy makes the result be non-Lagrangian.

To prove that any equation of motion is non-Lagrangian one should apply the Helmholtz operator to it. If the result is non-vanishing, the PDE is non-Lagrangian.

and moments later:

Okay, so the Helmholtz operator takes a PDE, incarnated as the differential form on the jet bundle which vanishes where the jets satisfy the PDE, to the formally anti-self-adjoint part of its evolutionary derivative. Unwinding this somewhat heavy variational calculus terminology in the simplistic case of a linear PDE comes down to saying that it must be formally self-adjoint in order to be variational. But the derivative in 2d along only one of two coordinates is clearly not.

This theory is so simple that it should help dispel any terror surrounding the phrase ‘QFT without a Lagrangian’, even if it’s of limited use in understanding the mysterious (2,0) theory.

As Urs earlier pointed out, if we study this theory on the cylinder $\mathbb{R} \times S^1$, it’s a special case of the chiral WZW model, adding:

For the abelian case this WZW model this is also called the “self-dual boson”, since one may interpret the chirality condition equivalently as saying that the “1-form field strength” is Hodge self-dual. From this perspective, the chiral WZW model is the first in a hierarchy of abelian self-dual higher gauge theories that exist in dimension $2+4k$.

For $k = 1$ this yields the abelian version of the 6d theory that we are talking about, with a 2-form gauge field $B$ (gerbe/2-bundle with connection) that has a self-dual 3-form field strength $H$.

Posted by: John Baez on December 29, 2017 12:10 AM | Permalink | Reply to this

### Re: An M5-Brane Model

Well, there are a few different reasons for why a ‘QFT without a Lagrangian’ would induce terror. the type you describe, where the theory is a summand of a lagrangian theory, invokes some minor terror (both chiral boson and self-dual gauge fields are of this type), however my personal terror of the 6d theory comes from the fact that it is always strongly coupled, and has no parameters.

Posted by: Ryan Mickler on December 29, 2017 5:22 AM | Permalink | Reply to this

### Re: An M5-Brane Model

it … has no parameters.

This is a general feature of ((1,0) or (2,0)) SCFTs in 6 dimensions (and $\mathcal{N}=3$ SCFTs in 4 dimensions): they have no relevant or marginal supersymmetry-preserving deformations.

The 5D SCFTs (and $\mathcal{N}\geq 3$ SCFTs in 3 dimensions) have no marginal deformations, though they do have relevant deformations.

In particular, that means that the 6D theories are not realizable as the IR fixed point of some asymptotically-free theory (which, back in the Stone Age, was the only way people knew how to think about conformal field theories).

Posted by: Jacques Distler on December 29, 2017 5:36 PM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

Hi John,

I was just browsing through the discussion when I came across your claim that the QFT of the “left-moving wave equation” $\partial_t \phi=\partial_x \phi$ has no lagrangian, which — or rather, its generalisations — is semi-common lore. As it happens the lore is untrue (modulo caveat involving the gauge transformation below) in this and other cases so I thought I might point to the lagrangian for this example here.

In fact the lagrangian for a theory whose solution space is spanned by $f(x+t)$ has been known since 1987 and was written down by Floreanini and Jackiw (formula (20) of R. Floreanini and R. Jackiw, Phys. Rev. Lett. 59, 1873 (1987).). That lagrangian is $\partial_x \phi (\partial_t \phi-\partial_x\phi)\,.$ The general solution to the equations of motion is $\phi(x,t)=\varepsilon(t)+f(x+t)$ however $\varepsilon(t)$ is pure gauge as $\delta_\varepsilon\phi=\varepsilon(t)$ is a gauge transformation leaving the lagrangian invariant up to a total derivative term. So, up to gauge transformations the solution space is the same as the one for the left-handed wave equation.

The keywords for the literature on this are chiral scalars'', orchiral p-forms” for the generalisation to a p-form field strength $F$ in 2p dimensions (for odd p) which is self-dual, i.e. $\star F=F$. For general p I think the original reference is M. Henneaux and C. Teitelboim, Dynamics of Chiral (Selfdual) P Forms, Phys. Lett. B {\bf 206}, 650 (1988).. A more recent, mathsier one is D.~Belov and G.~W.~Moore, “Holographic Action for the Self-Dual Field,” [hep-th/0605038]

Finally I should mention one can write down equations of motion for a field $\phi$ which cannot follow from any local lagrangian involving $\phi$ but which may actually follow from a lagrangian involving auxiliary fields. One example is in a paper I coauthored, “Yang-Mills as massive Chern-Simons theory: a third way to three-dimensional gauge theories”.

Best wishes, Alex Arvanitakis

Posted by: Alex Arvanitakis on February 9, 2018 10:18 PM | Permalink | Reply to this

### Re: An M5-Brane Model

I’m probably a little late to the party, but maybe my comments are still of interest.

First, a big thank-you to all of the interest in our work, and in particular to John Baez for even writing a blog post about this.

I’ll try to address the problems raised in the comments to the degree that I can and as soon as possible (the semester here is about to start and time is scarce), but already this much: We certainly do not claim to be experts in the intricacies of the (2,0)-theory. We just constructed a model that, after quite some efforts, finally manages to pass the consistency checks that are the most important/obvious ones to us. We certainly could have sat on this model and performed further checks; however, I think it’s more reasonable to say what we have and discuss with others. In particular, since our model joins building blocks that quite a few people and groups have been thinking about separately, from different directions.

Also, we still stand by the statements in the abstract. While our model is certainly not the full (2,0)-theory (at least the (1,0)-SUSY has to be completed, perhaps by non-local categorified monopole operators, to the full (2,0)-SUSY), we believe that it is pushing the current attempts at a classical description much further: This is an action of interacting tensor multiplets in a solid mathematical formulation allowing for global descriptions, which allows for a reasonable reduction mechanism to 4d SYM and 3d M2-brane models (if we forget about the issue of S-duality for the moment). From Jacques comments, it seems that our enthusiasm comes across as if we wanted to claim that we did more than stated in the abstract, which is not the case. We plan to put out an updated version of our paper very soon, where we’ll address this issue, if actually present.

Posted by: Christian Saemann on January 4, 2018 4:24 PM | Permalink | Reply to this

### Re: An M5-Brane Model

Christian - thank you for your comments. I should first say I apologize if this comes off combative, seems a feature of the medium. I find your proposal very intriguing, though I am unqualified to parse it in any detail - my knowledge of physics is very coarse and formal and non-physical (basically hearsay) - indeed one reason I love the (2,0) theory (or “theory $X$”, which I think is a more suggestive name) is my physicist friends assure me (thanks in particular to its purported non-Lagrangian nature) their understanding of it is also formal, so I feel like less of an impostor playing axiomatic games with it… but I am certainly rooting for you or someone to clarify what in the world this theory actually IS.

Let me try to pinpoint my confusion here. I’m happy calling the objects you’re considering built out of the string 2-group nonabelian gerbes. The issue is not if they’re abelian or not, but if they’re the RIGHT nonabelian object, and my feeling is that the sought-for nonabelian object doesn’t exist in the current language.

I think we can agree the difference between the Kac-Moody central extension and the loop group is important, but it’s not mysterious, and one can often ignore it at an informal level (or by saying words like “projective representation”). The difference between the string 2-group and a compact group $G$ is (in my understanding) completely analogous (in fact related by a looping), we have a $BU(1)$ central extension of $G$, whose (categorical) actions are “projectively” the same as $G$ actions. Clearly there’s some beautiful and involved math to make sense of bundles, connections etc in this setting but I think this is the gist of it.

Anyway the point is from what I’ve understood the object needed for the fields of the (2,0) theory is something different - it’s roughly a bundle for the group BG. This group doesn’t exist, but say in the abelian setting when G is a torus we need something like a BT bundle (or T-gerbe), not something like a bundle for a $BU(1)$ extension of T, which seems to me the natural abelian counterpart of a string connection. And indeed the Higgs mechanism for the nonabelian tensor field (or moving out on the Coulomb branch) produces exactly the (now perfectly understood) theory of T-gerbes.

So that’s my confusion - it seems to me the point of whether we centrally extend $G$ or not (in a higher sense) is not the problem, the problem is we need to somehow deloop $G$, and we can only do that for $G$ abelian. The string 2-group is clearly part of the story, as Fiorenza-Sati-Schreiber show, in the sense that the (2,0) theory should be holographically dual to a string 7d Chern-Simons theory — ok I don’t understand this sentence, but do understand a (probably closely related) one of Freed-Teleman where the (2,0) theory appears as a boundary condition for an understandable 7d TFT.

(I don’t understand the arguments about dimensional reduction but I’m told eg 6d SYM also reduces to 4d N=4, though without the manifest S-duality (geometric interpretation of the coupling constant as modulus of the compactification curve) that we get from the (2,0) theory.. is that what Jacques is saying?)

Posted by: David Ben-Zvi on January 5, 2018 3:49 AM | Permalink | Reply to this

### Re: An M5-Brane Model

So just as there is the groupoid principle bundle construction generalising ordinary principal bundles, one wants something similar for principal 2-bundles?

But there is already the notion of a groupoid-principal infinity-bundle.

Posted by: David Corfield on January 5, 2018 10:03 AM | Permalink | Reply to this

### Re: An M5-Brane Model

I think what you’re asking about relates to the notion of fiber bundle with fiber the space or $\infty$-groupoid $BG$ – that’s the same (I’m only thinking homotopically, not differential geometrically) as a principal bundle for the ($\infty$-)group $Aut(BG)$ (for a topological group $Aut(BG)$ is I believe a 2-group built out of $Out(G)$ and the center of $G$ and some “k-invariant” extension data). This is quite different than what we’d like, a principal bundle for a would-be group $BG$. In particular we would like some kind of higher groupish beast which has as a “maximal torus” the (well-defined) group $BT$, because one of the main things we know about this mysterious (2,0) theory is that whatever it is we can break its symmetry to make a theory of $BT$-bundles ($T$-gerbes).

Posted by: David Ben-Zvi on January 5, 2018 2:16 PM | Permalink | Reply to this

### Re: An M5-Brane Model

No need to apologize, I’m more than happy about any constructive criticism and the point you raise is certainly an important one.

You say that since the Coulomb branch for $n$ M5-branes is described by an abelian $U(1)^n$-gerbe, we would like to turn $U(1)^n$ into a non-abelian group (as happens for D-branes and $U(1)^n$-bundles), instead of having a single U(1) centrally extending a non-abelian group to the String-group, right?

As I said in the reply to Jacques’s post, we do not have the answer (yet), but for what it’s worth, here are my thoughts:

I agree that the above argument seems reasonable, and I believe that this is a point that many people have struggled with (including me). But the fact that we do not have the group BG for G non-abelian suggests that we may want something else. We note that the input into the (2,0)-model is an ADE-type singularity. From this, we get a Lie algebra, which we can uniquely extend to the string Lie 2-algebra and further, rather uniquely, to a metric Lie 3- and Lie 4-algebra.

For two M5-branes, our model is doing what one would want: we do have 2 tensor multiplets, and the reductions seem to work. Also, string(3), i.e. the Lie 2-algebra with underlying $\mathfrak{g}=\mathfrak{su}(2)$, is really the categorified analogue of $\mathfrak{su}(2)$ from many different perspecives. I think this is already quite good news.

The problems start for 3 M5-branes. Ideally, we would want to combine $n$ or $n-1$ copies of the string Lie 2-algebras of $\mathfrak{su}(n)$ in the $A_{n-1}$ case, which would reproduce the famous scaling of the degrees of freedom with $n^3$ for $n$ the number of M5-branes. The issue is now to deform these direct sums into something more interesting, i.e. appropriately interacting.

We are currently working on this and hope that our reduction to string theory together with the algebraic constraints we have give a satisfying answer.

Posted by: Christian Saemann on January 6, 2018 8:54 PM | Permalink | Reply to this

### Re: An M5-Brane Model

I agree with the Lorentzian signature problem. The original hope was that this can be solved similarly to the Lorentzian 3-algebra M2-brane models but at the moment, we have no solution to this.

It may not be obvious from the Lagrangian, but if you look at the equations of motion (4.64) and (4.65), then the tensor multiplet indexed with an “r” indeed decouples: the map $\nu_2$ sees only the one indexed with an “s”.

The $\mathfrak{u}(1)$-part of super Yang-Mills in 4d should certainly come from a reduction of some abelian tensor multiplet in 6d. This is what you can have in our model: Take the decoupled tensor multiplet $r$ and reduce it as usual.

But why are you so sure that the remaining $\mathfrak{su}(2)$ part should not arise from a Yang-Mills-like action in 6d? I would be very interested in understanding any such argument and grateful for some pointers/details.

Posted by: Christian Saemann on January 11, 2018 9:51 AM | Permalink | Reply to this

### Re: An M5-Brane Model

But why are you so sure that the remaining $\mathfrak{su}(2)$ part should not arise from a Yang-Mills-like action in 6d? I would be very interested in understanding any such argument and grateful for some pointers/details.

This is really important. It gets to the heart of why 4D $\mathcal{N}=4$ SYM has S-duality. I’ve said everything I’m about to say in bits and pieces in previous comments above. Evidently, the pieces haven’t all fit together for you, so let me go through it again, from the beginning.

Let us start in 6D. To make the physics clear, let us move out onto the tensor branch of the (2,0) theory. The low-energy physics involves $r$ tensor multiplets, and a collection of BPS strings, whose charges under the 1-form gauge symmetry are in 1-1 correspondence with the roots of $\mathfrak{g}$.

So far so good?

Now let us compactify the situation on a circle. We find ourselves at a point on the Coulomb branch (which is isomorphic to the tensor branch of the 6D theory) of a 5D SYM theory. The $r$ tensor multiplets become the gauge fields for the unbroken Cartan subgroup of the gauge group, and the W-bosons (the massive gauge fields, corresponding to the broken generators) are the aforementioned BPS strings, wrapped on the $S^1$.

Still with me?

Repeat this mantra to yourself, until you believe it in your bones: “The gauge bosons of the Cartan are the dimensional reduction of the 6D tensor multiplets; the massive W-bosons are the wrapped strings.”

OK. Now let’s compactify on an additional circle to get down to 4D. The 5D Yang-Mills fields reduce to 4D Yang-Mills fields, and we find ourselves, again, at a point of the Coulomb branch of 4D SYM.

Remember our Mantra. The gauge bosons for the Cartan subgroup are the dimensional reduction of the tensor multiplets from 6D. The massive W-bosons are wrapped strings.

But wait! Whoa! There are lots more ways to wrap a string around a $T^2$ than there were on $S^1$. What the hell is going on?

That’s correct. The strings wrapped around the first circle are W-bosons. The strings wrapped around the second circle are the ‘t Hooft-Polyakov monopoles of the 4D theory. Just like the W-bosons, the monopoles transform as part of a massive vector multiplet.

In fact, for any $(p,q)$-cycle on the torus, there are dyons which transform as part of massive vector multiplets.

(Technical remark: the massive vector multiplet of $\mathcal{N}=4$ is BPS. That means that the masses of the W-bosons and monopoles and dyons, which belong to massive vector multiplets, are protected from quantum corrections and can be computed exactly. They agree with those of the aforementioned wrapped BPS strings. The same remark also held for the massive (BPS) W-bosons in 5D.)

Now we understand S-duality. Exchanging the roles of the two circles inverts the gauge coupling and makes the erstwhile monopoles of the original gauge theory into the massive W-bosons of the new gauge theory (and vice-versa).

Now, repeat your new Mantra until you believe it in your bones: “Under S-duality, the Cartan subgroup is common to the two theories. The monopoles of the old theory are the W-bosons of the new theory and vice-versa. Both the monopoles and W-bosons originate from 6D as wrapped strings.”

All this was out on the Coulomb branch, where the physics is clear. As we approach the origin of the Coulomb branch, the W-bosons become massless, as do an infinite tower of monopoles and dyons. The physics at the origin is complicated. But it’s continuously connected to the physics we (now, hopefully) understand.

Now that you understand how the S-duality of the 4D $\mathcal{N}=4$ SYM is intimately tied up with how the 4D $\mathcal{N}=4$ SYM is realized from the $T^2$ compactification of the 6D (2,0) theory, I think you can answer your own question.

Posted by: Jacques Distler on January 11, 2018 6:57 PM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

But why are you so sure that the remaining $\mathfrak{su}(2)$ part should not arise from a Yang-Mills-like action in 6d?

Or I could have been more succinct and reiterated that (2,0) supersymmetry has only one massless represention – the tensor multiplet. It (unlike (1,0) supersymmetry) does not admit a vector multiplet.

But I think the more long-winded explanation of the correct answer is more illuminating.

Posted by: Jacques Distler on January 12, 2018 3:22 AM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

Thanks for your comments, they are great. However, please allow me to push my arguments a little more.

I think I understand the Coulomb branch reasonably well and fully agree with what you say. However, we already know that the abelian action of self-dual 3-forms does not allow for a continuous deformation to something non-abelian from the no-go theorem by arXiv:hep-th/9909094. This is reflected in our action by a Chern-Simons-like term, requiring a discrete coupling constant. All this suggests to me that the physics of the Coulomb branch and the interacting branch is in fact not connected continuously, at least not in 6d. Is there any argument for the continuity of physics here? Together with arXiv:hep-th/9909094, this could immediately rule out a classical action, and we could drop all other no-go arguments.

I’m also concerned about the succinct argument: Sure, there is no (2,0)-vector multiplet. However, we also have no general $\mathcal{N}=8$ M2-brane model. Here, we only get $\mathcal{N}=6$ (essentially the analogue of getting $\mathcal{N}=(1,0)$ in 6d) and we do have an additional, unexpected gauge potential. Important here is the Chern-Simons action, so that we do not add new degrees of freedom. We do have a higher Chern-Simons term in our action, so the analogy is actually quite close, even if it’s not quite perfect. The full $\mathcal{N}=8$ SUSY in the M2-brane model is restored from $\mathcal{N}=6$ by non-local monopole operators, the same could, conceivably, happen for an $\mathcal{N}=(1,0)$ model from non-local self-dual string operators, couldn’t it?

Posted by: Christian Saemann on January 12, 2018 10:19 AM | Permalink | Reply to this

### Re: An M5-Brane Model

I think I understand the Coulomb branch reasonably well…

I’m not sure you do.

However, … All this suggests to me that the physics of the Coulomb branch and the interacting branch is in fact not connected continuously, at least not in 6d.

You seem to be under the misapprehension that the theory on the Coulomb/Tensor branch is free. That is obvious nonsense.

If you took a free field theory and compactified it on a torus, it would not magically become an interacting theory.

The (2,0) theory, on its Coulomb/Tensor branch is IR-free (like QED or, more appositely, like $\mathcal{N}=4$ SYM on its Coulomb branch).

The UV physics is that of the (2,0) SCFT. The IR physics is weakly-interacting (free tensor multiplets + small corrections), in exactly the same way that the IR physics of QED is weakly-interacting (free Maxwell theory + small corrections).

If you work at a fixed energy, and tune the VEV $\langle\Phi\rangle\to0$, the physics approaches arbitrarily closely that of the (2,0) SCFT at the origin.

Is there any argument for the continuity of physics here?

Yes. But, first of all, you need to understand how the continuity could fail. There are examples of QFTs where the moduli space of vacua appears to be connected classically, but is disconnected quantum-mechanically. The “point” at which the two branches of the moduli space meet, which was at finite distance in field space (classically), turns out to be at infinite distance in the quantum-corrected metric on the field space.

That doesn’t happen here. The theory on the Coulomb/Tensor branch is a theory where the superconformal invariance is spontaneously-broken (the radial mode of $\Phi$ is a Goldstone boson). And there is enough supersymmetry to exactly determine the metric on the moduli space. And, yes, in the exact moduli space metric, the origin is at finite distance.

Together with arXiv:hep-th/9909094, this could immediately rule out a classical action, and we could drop all other no-go arguments.

I’m also concerned about the succinct argument: Sure, there is no (2,0)-vector multiplet. However, …

The statement is not about Lagrangians. The statement is about representation theory. It doesn’t matter whether some of the particles in the multiplet are created by fundamental fields in the theory and others by the analogue of monopole operators. The question is whether the representation in question exists.

Every representation of 3D $\mathcal{N}=8$ supersymmetry can be decomposed under $\mathcal{N}=6$. Similarly, every representation of 6D (2,0) supersymmetry can be decomposed under (1,0).

There are 3 massless representations$^\dagger$ of global (1,0) SUSY: the tensor multiplet, the hypermultiplet and the vector multiplet.

The (2,0) tensor multiplet (the only massless representation of (2,0) SUSY) decomposes as a (1,0) tensor multiplet $\oplus$ a (1,0) hypermultiplet. But there is no representation of global (2,0) SUSY which contains the (1,0) vector multiplet.

$^\dagger$ More precisely, these are all the representations with $j_1+j_2\leq 1$ (i.e., admissible in an interacting, non-gravitational, theory).

Posted by: Jacques Distler on January 12, 2018 5:23 PM | Permalink | Reply to this

### Re: An M5-Brane Model

However, we already know that the abelian action of self-dual 3-forms does not allow for a continuous deformation to something non-abelian … This is reflected in our action by a Chern-Simons-like term, requiring a discrete coupling constant. All this suggests to me that the physics of the Coulomb branch and the interacting branch is in fact not connected continuously, at least not in 6d.

Perhaps the source of confusion is what is meant by the word “branch” here. We are not talking about a family of different QFTs (obtained, say, by varying some coupling constant(s)). We are talking about a family of different vacua of the same QFT.

Your model may not have any adjustable coupling constants (indeed, it can be shown rigourously that a (1,0) or (2,0) SCFT has no relevant or marginal SUSY-preserving deformations), but it does have a moduli space of vacua. Unfortunately, it’s not the same moduli space of vacua as that of the (2,0) theory.

Posted by: Jacques Distler on January 13, 2018 9:22 PM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

Sorry for the long delay, my time is quite constrained at the moment.

I clearly have some more reading to do regarding the Coulomb branch. A few observations so far: Our model seems to be the SUSY- and PST-completion of the action mentioned e.g. in arXiv:1608.03919, equation (2.1). Our completion, however, links $F$ to its non-dynamical 4-form dual $G$. So our Yang-Mills part is much more restricted than the purely bosonic form given in this paper.

As far as the moduli space is concerned, one needs to be careful here: We could have formulated our model using the equivalent loop space model of the string Lie 2-algebra given in arXiv:math.QA/0504123. This would look naively as if we had infinitely many tensor multiplets, as $B_r$ and $B_s$ now take values in the Lie algebra of based loops $\hat \Omega \mathfrak{g}$. Since this gauge Lie 2-algebras is categorically equivalent to the skeletal one we used, one would expect that the physics is the same. We explained that and how this works for the BPS subsector in arXiv:1705.02353, and we’ll add another section concerning this to version 2 of our last paper. So the notion of isomorphism between moduli spaces for higher gauge theories is much more involved.

Finally, my argument that supermultiplet analysis is not enough seems to have been unclear: In the case of the BLG M2-brane model, we take the N=8 multiplet with values in some vector space. We then add a gauge potential (NOT another N=8 multiplet) without propagating dofs due to Chern-Simons eom. Just looking at the possible supermultiplets in 3d, we would have missed this possibility. The 6d analogue is to take the field content of the (2,0)-tensor multiplet and add some gauge and higher potentials, which do not propagate due to higher CS equations of motion. This is roughly what we have in our model.

Posted by: Christian Saemann on January 19, 2018 11:01 AM | Permalink | Reply to this

### Re: An M5-Brane Model

Our completion, however, links $F$ to its non-dynamical 4-form dual $G$. So our Yang-Mills part is much more restricted than the purely bosonic form given in this paper.

Maybe you need to explain that.

Your Lagrangian contains $L = \dots + \phi Tr F^2 +\dots$ When $\phi$ has a large VEV, this leads to weakly-coupled propagating Yang-Mills gauge bosons. Indeed, you used this in arguing for the emergence of 4D Yang-Mills, upon compactification.

Is the above paragraph incorrect, either because

• $\phi$ cannot be given a nonzero VEV, or
• something in the $\dots$ renders the Yang-Mills fields non-dynamical, despite appearances to the contrary?

Clearly, we need to settle this, before anything else. Does your model, at large $\langle\phi\rangle$, have propagating Yang-Mills gauge bosons, or doesn’t it?

As far as the moduli space is concerned, one needs to be careful here: We could have formulated our model using the equivalent loop space model of the string Lie 2-algebra given in arXiv:math.QA/0504123. This would look naively as if we had infinitely many tensor multiplets, … categorically equivalent….

Sorry. I absolutely don’t understand.

If the theory is on its Coulomb/Tensor branch, the physics is IR-free, and a 6D physicist can do an experiment to count the number of propagating degrees of freedom. That number is an invariant of the theory; it can’t depend on what formulation you use.

Moreover, the 1-particle states must fall into representations of (2,0) supersymmetry. And there is only one massless irreducible representation – the (2,0) tensor multiplet.

If your model (for large $\langle\phi\rangle$) has propagating Yang-Mills gauge bosons, then it can’t possibly be (2,0) supersymmetric.

And, in either case, there can’t possibly be any ambiguity as to the number of propagating self-dual tensors.

Finally, my argument that supermultiplet analysis is not enough seems to have been unclear: In the case of the BLG M2-brane model, we take the N=8 multiplet with values in some vector space. We then add a gauge potential (NOT another N=8 multiplet) without propagating dofs due to Chern-Simons eom. Just looking at the possible supermultiplets in 3d, we would have missed this possibility.

I am kinda familiar with the BLG model, and I disagree with your characterization of it.

But we don’t need to discuss the physics at the origin of the moduli space of vacua. The IR physics, way out on the Coulomb branch is a perfectly conventional theory of massless abelian vector multiplets with manifest $\mathcal{N}=8$ supersymmetry. We are discussing the analogue of that situation here.

Posted by: Jacques Distler on January 19, 2018 8:07 PM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

Since the world-volume theory on the M2-brane seems to be your “guiding light” for the M5-brane case, let me say a few things about that theory which would be clarifying.

The theory on a stack of $N$ (coincident) parallel M2-branes is the infrared fixed point of $\mathcal{N}=8$ $U(N)$ SYM in 3 dimensions.

This theory, like its maximally-supersymmetric cousin in 4D, has a Coulomb branch of vacua, along which the physics is IR-free — $N$ massless abelian vector multiplets.

BLG found a formulation of the theory of two M2-branes transverse to a certain orbifold. ABJM generalized this to $N$ M2-branes. Far out on the Coulomb branch, in either formulation, the physics approaches that of M2-branes in flat space.

For $k=1,2$, the theory is fully $\mathcal{N}=8$ supersymmetric; for higher Chern-Simons level, the supersymmetry is only $\mathcal{N}=6$. So, far out on the Coulomb branch, the massless spectrum is ($\mathcal{N}=8$)-supersymmetric, but the interactions are not.

You have a different problem: the massless spectrum of your model, far out on its moduli space of vacua, is not (2,0)-supersymmetric. So it hardly matters that the interactions also violate (2,0) supersymmetry.

Posted by: Jacques Distler on January 20, 2018 6:10 AM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

Concerning the Yang-Mills part: Our PST completion of the action (which is the SUSY extension of that of arXiv:1305.1304) gives the following two equations of motion (4.64): self-duality of the 3-form curvature (mildly corrected by a fermionic term) as well as $G = \star \phi \wedge F$ up to fermionic corrections, where G is the curvature of a non-dynamial 3-form $C$. If you give $\phi_{s}$ a large expectation value, then the term $\langle \Phi\iota_V \star G),\star(\iota_V * G)\rangle$ in (4.45) drops out ($\Phi(-)$ is proportional to $\frac{1}{\phi_{s}}$). So to leading order, you just get Yang-Mills.

Concerning categorical equivalence: Sorry, I thought I’d seen you discuss string Lie 2-algebras before. If you consider a higher gauge theory on a principal 2-bundle (let’s restrict to this for now), then you have a categorified gauge group and a resulting gauge Lie 2-algebra. One easy formulation of such as Lie 2-algebra is as a 2-term $L_\infty$-algebra, which is a pair of vector spaces $V\rightarrow W$ with totally antisymmetric brackets $\mu_1$, $\mu_2$ and $\mu_3$ with 1, 2 and 3 slots satisfying some generalized Jacobi identities. Essentially, $\mu_1$ is a differential, $\mu_2$ is a generalized Lie bracket and $\mu_1\circ\mu_3$ encodes the failure of the Jacobi identity. Just as we expect gauge theories with isomorphic Lie algebras to describe the same physics, I believe it’s not too far fetched to assume that higher gauge theories with isomorphic $L_\infty$-algebras also describe the same physics. This is necesssary, at least, if the higher parallel transport etc., are all well defined. Now the question is what the correct notion of isomorphism between $L_\infty$-algebras is. Everybody who looked at this properly (and I know that Urs agrees) concludes that these are not naive isomorphisms, but quasi-isomorphisms. They are more general than the obvious chain isomorphisms, and as explained in arXiv:math.QA/050412 they show that the Lie 2-algebras $\hat \Omega \mathfrak{g}\rightarrow P_0\mathfrak{g}$ and $\mathbb{R}\rightarrow \mathfrak{g}$ are quasi-isomorphic. So mathematically, we would assume that the physics based on either of these should be equivalent. We actually showed in arXiv:1705.02353 that this works in at least one case: solutions and gauge orbits of the BPS equations of our model map nicely into each other. We’ll add a section concerning this in v2 of our paper.

I appreciate that these quasi-isomorphism sound at first fantastic because the dimensions of the involved vector spaces don’t have to match. But this is just because there are higher gauge symmetries around that introduce redundancies which can be used to reduce the dimensions again. I’m sure that you actually encountered these quasi-isomorphisms before (possibly not consciously): In BV-quantization, you add “trivial pairs” when gauge fixing. This is precisely switching to a quasi-isomorphic BV-complex. And this is possibly a nice example that just looking at the lowest dimensional example of the string Lie 2-algebra to identify the moduli space is not a good idea. If we did this in BV, we couldn’t gauge fix since the gauge fixing fermion doesn’t exist. If I would allow myself to speculate wildly, then I would say that these quasi-isomorphisms in the 6d theory might actually have a tiny chance of explaining some of the crazy aspects of S-duality via different higher gauge fixing. But clearly our model is already controversial as it is, so let’s not go there.

Altogether, you can count bosonic and fermionic degrees of freedom but which of these are 1-gauge and which of these are 2-gauge or tensor fields might be changed by quasi-isomorphisms of the gauge structure. Again, I don’t claim that I understand this fully, but I don’t think anybody else does, either. (I’d expect Urs to have some deeper thoughts on this; but at least it’s not developed in the literature.)

Concerning the M2-brane model: I’m quite surprised that you don’t like my interpretation of the field content of this Lagrangian (but that’s great, then I can possibly learn something here). So let’s forget about moduli spaces of the QFT, IR-flows and even M2-branes and just look at classical kinematical data. I think then my characterization is correct: The field content of the M2-brane doesn’t form any N=8 multiplet you would have obtained by simply classifying representations of the 3d SUSY algebras. You start from the the usual N=8 multiplet used in the description of a single M2-brane and then you add by hand a Lie algebra valued gauge potential to render the interactions SUSY invariant.

Before adding the (1,0)-vector multiplet our model is non-interacting and has the same field content as the (2,0)-model (since we have a (1,0)-tensor multiplet and an appropriate (1,0)-hypermultiplet). At this point, we are exactly in the same situation as in the case of the non-interacting M2-brane model. Now we want to turn on interactions, which we can’t do without new gauge fields (since the BU(1)^n-gerbes cannot be turned into BU(n)-gerbes, as the latter don’t exist). I agree that there’s no particular motivation for these additional fields to form a (1,0)-multiplet, this is just because we started from the model of arXiv:1108.4060. Indeed, we should possible try to drop this assumption to overcome some of our other problems. Anyway, roughly following the construction of the M2-brane model, we introduce a Chern-Simons type interaction and PST terms to fix F in terms of other degrees of freedom. And we do all this in the unique, mathematically consistent fashion that preserves N=(1,0) SUSY, the analogue of the N=6 SUSY of the general M2-brane model.

Posted by: Christian Saemann on January 20, 2018 11:17 AM | Permalink | Reply to this

### Re: An M5-Brane Model

I am well-aware that in cohomological field theories (probably not the right generic term, here, but I will use it anyway) the “naïve” degrees of freedom, that one sees in the Lagrangian, do not generate the true physical spectrum; the latter is obtained by passing to the cohomology of some complex.

• they are in the physical spectrum (in which case, your model cannot possibly be (2,0) supersymmetric) or
• they are eliminated from the physical spectrum (in which case, they are not available to give rise to 4D Yang-Mills, upon compactification – which I thought was the main “result” of your paper).

Which is it?

The same remarks apply, mutatis-mutandis, to the tensor multiplets in your model. I don’t know what it means for two quantum field theories to be “categorically-equivalent.” But, in my limited experience, passing to the physical spectrum, from the naïve one, doesn’t add new degrees of freedom; it removes them.

So, since the model in your paper has two tensor fields, I assume that the true number of physical tensor multiplets is at most two (not $\text{rank}(\mathfrak{g})$, as it would be if the theory with gauge algebra $\mathfrak{g}$ were to describe the type-$\mathfrak{g}$ (2,0) theory).

Moreover, because of the indefinite sign of their kinetic terms (the signature is (1,1)), the theory would be non-unitary, unless at least one of them is removed from the physical spectrum.

As to the M2-brane case, all I can say is that you are confused. The theory of $N$ M2-branes in flat space has manifest $\mathcal{N}=8$ supersymmetry. The low-energy effective field theory on the Coulomb-branch consists of $N$ abelian $\mathcal{N}=8$ vector multiplets, and the interactions are $\mathcal{N}=8$ supersymmetric.

The ABJM model (generalizing BLG) describes $N$ M2-branes on a $\mathbb{Z}_k$ orbifold. Again, the degrees of freedom on the Coulomb branch consist of $N$ abelian $\mathcal{N}=8$ vector multiplets. But the interactions (at least for $k\gt 2$) preserve only $\mathcal{N}=6$ supersymmetry. This is not some defect of the presentation. The true theory doesn’t have $\mathcal{N}=8$; it has only $\mathcal{N}=6$.

Your problem is that – if the Yang-Mills degrees of freedom are physical – the degrees of freedom on the tensor branch of your theory are incompatible with (2,0) supersymmetry. Since the theory is IR-free, the interactions cannot possibly save you.

Posted by: Jacques Distler on January 21, 2018 7:27 PM | Permalink | PGP Sig | Reply to this

### Re: An M5-Brane Model

In our model, as it’s written down at the moment, the Yang-Mills degrees of freedom are fixed by the four-form curvature $\mathcal{G}$ but there is no further equations on either the 2-form or 4-form curvatures. So you can say that the YM degrees of freedom are there and some mechanism, presumably for the 4-form field strength, is needed to eliminate them.

(BTW: In my opinion, the reduction is not the main result; this was just a nice side effect. Our main result is that the String Lie 2-algebra gives rise to a suitable gauge structure of the model of Samtleben et al. and that you can then complete the action via a PST mechanism, even including hypermultiplets. That is, that this model is a proper higher gauge theory, therefore is mathematically consistent as a classical gauge theory on general spacetimes, and matches various other expectations, in particular the appearance of the string Lie 2-algebra. Secondary observations are then the dimensional reductions to Yang-Mills and Chern-Simons matter, which I think are nice.)

Your experience with categorical equivalence has misled you here: The string Lie 2-algebra model we used in the paper is the minimal one, as this 2-term sequence is the cohomology of the 2-term sequences quasi-isomorphic to the string Lie 2-algebra. All other, equivalent models will have more degrees of freedom. That was my point. As stated before, I certainly agree with the problem of the indefinite signature term.

For the M2-brane, I find it interesting how hard it seems to convey a simple argument via a comment on a blog. (And I highly doubt that there’s confusion on my part on this point I was making.)

I agree that if the Yang-Mills degrees of freedom are physical, as they are up to duality with the four-form curvature, then (1,0) supersymmetry cannot be enhanced to (2,0). However, a mechanism that renders these topological would complete my analogy with the M2-brane model, which I think is a valid and valuable one.

Posted by: Christian Saemann on January 22, 2018 10:27 AM | Permalink | Reply to this

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