### 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

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

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):

As Chethan points out in the comments, it’s never a good time to try to write about this stuff.In Monday’s listings, a paper by Aharony *et al* appeared. They couple the standard $\mathcal{N}=3$ supersymmetric Chern-Simons theory^{1} 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

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

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 $S_{\text{matter}}^{\mathcal{N}=3} = S_{Q} + S_{\tilde{Q}} + \int d^3 x \left(\int d^2\theta \tilde{Q} \Phi Q + \text{c.c.}\right)$

## Re: Bagger-Lambert Again

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