The n-Category Café

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:

• André Henriques, Integrating $L_\infty$-algebras.

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.

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’.

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

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.)

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).

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?

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.

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…

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

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 $H$.

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?)

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.

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.