## June 10, 2008

### Bagger-Lambert Again

I haven’t attempted to post more about Bagger-Lambert Theory, since my earlier post. Every time I think it might be worthwhile to pause and take stock of developments, two or three new papers on the subject appear on the arXivs, and I drop that silly idea.

Still, one thread which got a fair amount of attention was the proposal by three different groups of a whole new class of “Bagger-Lambert” algebras, obtained by relaxing the condition that the bilinear form (the “trace”) on the algebra be positive-definite.

With an indefinite bilinear form, it sure looks like the theory has ghosts. Which, to put it mildly, would not be good. A way around this difficulty was proposed by two other groups: by gauging a certain shift symmetry, one can remove the negative-norm states.

Well, along come Ezhuthachan, Mukhi and Papageorgakis, who point out that the resulting theory is on-shell equivalent to the standard $D=3$, $\mathcal{N}=8$ SYM — that is (for one of the classical gauge groups) to the theory on the world volume of a stack of D2 branes. In this dictionary, there is a scalar field, whose VEV is the Yang-Mills gauge coupling. For any finite value of the VEV of that scalar, the would-be $SO(8)$ R-symmetry is broken to $SO(7)$ (as is the superconformal symmetry).

The computation involves a nice application of a nonabelian duality transformation, due to de Wit, Nicolai and Samtleben. Consider

(1)$\int d^3 x \tfrac{1}{2} Tr\left(\epsilon^{\mu\nu\lambda} B_\mu F_{\nu\lambda} - {(D_\mu\phi - g_{\text{YM}}B_\mu)}^2\right)$

In addition to the usual Yang-Mills gauge transformations, $\begin{gathered} A_\mu \to g A_\mu g^{-1} + \partial_\mu g g^{-1}\\ \phi \to g \phi g^{-1},\quad B_\mu\to g B_\mu g^{-1} \end{gathered}$ (1) is invariant under a local shift symmetry

(2)$\phi \to \phi + g_{\text{YM}}\chi,\quad B_\mu\to B_\mu + D_\mu \chi$

Using (2) to gauge away $\phi$, $B_\mu$ becomes an ordinary auxiliary field, and (1) is on-shell equivalent to $\int d^3 x\, -\tfrac{1}{4 g_{\text{YM}}^2} Tr(F_{\mu\nu}^2)$

Now, the observation of Ezhuthachan et al is that $\phi$ looks like an eighth adjoint-valued scalar, to complement the seven already present in $\mathcal{N}=8$ SYM. Its coupling to $B_\mu$, however, breaks that symmetry. To restore the symmetry, treat $g_{\text{YM}}$ as the VEV of another ($SO(8)$ vector-valued) scalar, $\langle Y\rangle = (0, 0, 0, 0, 0, 0, 0, g_{\text{YM}})$. To ensure that $Y^I$ is spatially-constant, they introduce a vector field, $C^I_\mu$ (and another scalar, $Z^I$), with action $\int d^3 x\, (C^I_\mu - \partial_\mu Z^I) \partial^\mu Y^I$ and local shift symmetry $Z^I \to Z^I + \eta^I,\quad C^I_\mu \to C^I_\mu + \partial_\mu \eta^I$

Putting all the pieces together, they show that this construction yields the gauged version of the “new” Bagger-Lambert actions. Thus, with some particular choice of VEV for $Y^I$, the latter is just on-shell equivalent to standard $\mathcal{N}=8$ SYM.

That was fun while it lasted …

#### Update (6/11/2008):

In Monday’s listings, a paper by Aharony et al appeared. They couple the standard $\mathcal{N}=3$ supersymmetric Chern-Simons theory1 to matter. For a particular choice of gauge group — $U(N)\times U(N)$ or $SU(N)\times SU(N)$, with Chern-Simons levels $(k,-k)$ — and matter representations — chiral multiplets $A_i\in(N,\overline{N})$ and $B_i\in (\overline{N},N)$, $i=1,2$ — they show that the resulting theory has an enhanced $\mathcal{N}=6$ supersymmetry. The $SO(3)_R$ symmetry is enhanced to and $SO(6)_R$, under which the scalars $(A_1,A_2,\overline{B}_1,\overline{B}_2)$ transform as a $\mathbf{4}$. In the particular case when the gauge group is $SU(2)\times SU(2)$, the $(2,2)$ is a real representation, and the $SO(6)_R$ is enhanced to an $SO(8)_R$.

For higher $N$, they argue that the theory (with $\mathcal{N}=6$ superconformal invariance) is the correct description of $N$ M2-branes transverse to a $\mathbb{C}^4/\mathbb{Z}_k$ orbifold. There is much here that bears further discussion. Perhaps fodder for another post …

1 The $\mathcal{N}=2$ supersymmetric Chern-Simons action is

(3)\begin{aligned} S_{\text{CS}}^{\mathcal{N}=2} &= \tfrac{k}{2\pi}\int d^3x \int d^4\theta \int_0^1 d\tau Tr\left(V \overline{D}^\alpha \left(e^{-\tau V} D_\alpha e^{\tau V}\right)\right)\\ &= \tfrac{k}{4\pi} \int A d A +\tfrac{2}{3} A^3 - \overline{\chi}\chi + 2 \sigma D \end{aligned}

in Wess-Zumino gauge. Here $D$ and $\sigma$ are scalar fields, and $\chi$ is a Dirac fermion, all in the adjoint. The $\mathcal{N}=3$ Chern-Simons action contains an additional chiral multiplet, $\Phi$, in the adjoint

(4)$S_{\text{CS}}^{\mathcal{N}=3} = S_{\text{CS}}^{\mathcal{N}=2} - \tfrac{k}{4\pi}\int d^3 x \left(\int d^2\theta Tr(\Phi^2) + \text{c.c.}\right)$

We can couple matter chiral multiplet(s), in representation $R$, to (3) \begin{aligned} S_{Q} &=\int d^3x \int d^4\theta \overline{Q}e^V Q\\ &= \int d^3x \overline{D_\mu Q} D^\mu Q + i \overline{\psi} \gamma^\mu D_\mu \psi\\ & +\overline{Q}(D-\sigma^2)Q - \overline{\psi}\sigma\psi + i\overline{Q}\overline{\chi}\psi - i\overline{\psi}\chi Q \end{aligned} To get $\mathcal{N}=3$ supersymmetry, the matter $(Q,\tilde{Q})\in R\oplus \overline{R}$, and the action $S_{\text{matter}}^{\mathcal{N}=3} = S_{Q} + S_{\tilde{Q}} + \int d^3 x \left(\int d^2\theta \tilde{Q} \Phi Q + \text{c.c.}\right)$

Posted by distler at June 10, 2008 11:30 PM

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### Re: Bagger-Lambert Again

That was fun while it lasted…

So what is the lesson to be learned here? Just a weird reformulation of something known, or a reformulation of intrinsic value?

Posted by: Urs Schreiber on June 11, 2008 3:33 AM | Permalink | Reply to this

### Re: Bagger-Lambert Again

So what is the lesson to be learned here?

I would say: “Naïve attempts to generalize the Bagger-Lambert construction fail.”

• Relaxing the positivity constraint on the bilinear form allows for new solutions (one for any compact Lie algebra, $\mathfrak{g}$), but these new theories appear not to be unitary.
• A suitable gauging of the result leads to a unitary theory: the only unitary theory you know about with this amount of supersymmetry, the $\mathcal{N}=8$ SYM with gauge group $G$.

More broadly, I would say the lesson is: “Physics is hard.” But it’s not clear how one ought to modify one’s behaviour to take that lesson into account.

Posted by: Jacques Distler on June 11, 2008 8:20 AM | Permalink | PGP Sig | Reply to this

### Re: Bagger-Lambert Again

> That was fun while it lasted …

Dear Jacques, that sounds depressing, but I am not sure the story is that bleak. I am sure you have seen the paper by Aharony et al. 0806.1218, generalizing the original Bagger-Lambert without getting caught up in the indefinite trace form. Unfortunately, their theory has only 6 SUSYs manifest in the general case. But when the gauge group is SU(2) * SU(2) there is an enhancement in SUSY and you end up with van Raamsdonk’s rewriting of the original Bagger-Lambert. So the 3-algebra structure in this picture is almost incidental.

Perhaps the message is only that 3-algebras might not be the right way to go about constructing/generalizing M2-brane theories.

Warm regards,
Chethan.

Posted by: chethan krishnan on June 11, 2008 10:56 AM | Permalink | Reply to this

### Re: Bagger-Lambert Again

the message is only that 3-algebras might not be the right way

Jim Stasheff emphasized that what lately is being called “3-algebra” is precisely a Nambu bracket.

Nambu brackets have been considered in the study of membranes before, but in a context that is at least superficially a bit different. Do any of the recent 3-algebra articles comment on that?

Posted by: Urs Schreiber on June 11, 2008 11:21 AM | Permalink | Reply to this

### Re: Bagger-Lambert Again

Matsuo and co. have papers on explicit constructions of 3-algebras using the Nambu-Posisson bracket of an algebra of functions on a manifold. But these things are infinite dimensional and were used by them for generating M5s from the M2s (among other things). In a “normal” Bagger-Lambert theory we are looking for something much more like the gauge structure constants of a Yang-Mills theory, so that we can, for example, relate it to D2s by giving VEVs to some transverse scalars.

I am totally clueless about the pre-Bagger-Lambert work on the relation between M2-branes and 3-algebras, but you should find references in Matsuo et al’s papers. Especially if this older work that you talk about was based on attempts to generalize D-brane noncommutativity to M-branes by messing with function spaces over manifolds or something like that.

Posted by: chethan krishnan on June 11, 2008 12:11 PM | Permalink | Reply to this

### Re: Bagger-Lambert Again

Thanks, that’s useful information.

Concerning that other occurence of Nambu brackets:

I think Nambu brackets were at some time hoped to yield a “covariant” matrix model description of the membrane, in that it would allow to write the action (as opposed to just the Hamiltonian) as a square of brackets, but now of trinary brackets.

So it’s a rather different context in which the trinary thing appears, and I don’t know if it every worked out. But I wouldn’t be surprised if there is a relation.

Posted by: Urs Schreiber on June 11, 2008 12:55 PM | Permalink | Reply to this
Read the post Lie 3-Algebras on the Membrane (?)
Weblog: The n-Category Café
Excerpt: Recently a trinary bracket appears in the study of the supermembrane which is sometimes addressed as a homotopy algebraic structure.
Tracked: November 6, 2008 7:22 PM

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