### Bagger-Lambert

The low-energy theory on a (stack of) D2-brane(s) is a maximally-supersymmetric gauge theory in 2+1 dimensions. The Yang-Mill multiplet has a gauge field and 7 real scalars in the adjoint representation. At least if you are out on the Coulomb branch, where the gauge symmetry is Higgsed down to the Cartan, you can dualize the gauge fields to another scalar, which is *circle-valued*.

The M2-brane is obtained as the strong-coupling limit of this theory. The radius of the circles (one for each element of the Cartan subalgebra) go to infinity, and the $SO(7)$ R-symmetry is promoted to $SO(8)$. This strong-coupling limit is superconformal, but the above description is effective only for the free theory, where the M2-branes are separated (away from the origin, the moduli space looks like $\mathbb{R}^{8n}/S_n$). The theory of coincident M2-branes is an interacting SCFT which, so far, does not have a Lagrangian description. But there’s no *theorem* that rules out a Lagrangian description, so there may just be one.

Bagger and Lambert recently proposed a very interesting maximally supersymmetric *interacting* 2+1D Lagrangian field theory which, at least classically, seems to be superconformal. It does not arise as the dimensional reduction of some higher dimensional theory, and so it was missed in previous attempts at tackling this problem.

I never got around to blogging about Bagger and Lambert’s paper, but Bandres, Lipstein and Schwarz wrote a nice followup, which gives me an excuse to return to the subject.

#### Update:

Whoops! Even as was typing this, Mark van Raamsdonk came out with a paper making some of the points below. I’d better hurry up and post this, before there are yet-more followup papers to discuss.Let me start with some notational preliminaries. The connected part of the Lorentz group in 2+1 dimensions is $SL(2,\mathbb{R})$, so we’ll use a Wess and Bagger-like notation for spinors^{1}. Spinor indices are raised and lowered using the 2-component $\epsilon$-symbol, $\epsilon_{\alpha\beta}= -\epsilon_{\beta\alpha}$. The $\gamma$-matrices are $\sigma^\mu_{\alpha\beta} = \sigma^\mu_{\beta \alpha}$, and are *real*. After raising an index using $\epsilon^{\alpha\beta}$, we can write $\tensor{(\gamma^\mu)}{^\alpha_\beta}= \epsilon^{\alpha\delta}\sigma^\mu_{\delta\beta}$ in terms of Pauli matrices, with $\gamma^0 =i\sigma_2$.

Also real are the $SO(8)$ $\gamma$-matrices, which we can regard as a trilinear form $\Gamma: V\otimes S\otimes C \to \mathbb{R}$ where $V$, $S$, and $C$ are the three 8-dimensional real representations of $Spin(8)$. Alternatively, letting “$I$” be an $SO(8)$ vector index, we think of $\Gamma^I: C\to S$ We’ll also need $\begin{aligned} \Gamma^{I J} &= \tilde{\Gamma}^{[I}\Gamma^{J]}: C\to C\\ \Gamma^{I J K} &= \Gamma^{[I} \tilde{\Gamma}^J \Gamma^{K]}: C\to S \end{aligned}$ where $\tilde{\Gamma}^I ={\left(\Gamma^I\right)}^t: S\to C$.

What Bagger and Lambert do is introduce an auxiliary vector space, $W$ (really, a vector bundle, which we will take to be trivial). $W$ is endowed with a positive definite inner product, $(\cdot,\cdot): \Sym^2(W)\to \mathbb{R}$ and a skew-symmetric quadrilinear form, $f: \wedge^4 W \to \mathbb{R}$ Alternatively, using the metric, we can regard $f$ as a trilinear product

The fields of the model are $\begin{aligned} \phi& \text{a scalar field taking values in}\, W\otimes V\\ \psi_\alpha& \text{a spinor field taking values in}\, W\otimes C\\ A_\mu& \text{a}\, G\subset SO(W)\, \text{gauge connection}\\ \end{aligned}$

Now, $so(W)\simeq \wedge^2 W$ has the usual action on $W$,

But, because we have (1), we can contemplate an “exotic” action,

We demand that $f$ be invariant under this action, which amounts to requiring

Then the antisymmetry of $f$ ensures that the inner product, $(\cdot,\cdot)$ is also invariant.

Unfortunately, (3) is not really satisfactory. The action of $so(W)$ must satisfy the the Lie-algebra relations,

so it would make sense, for instance, to gauge that symmetry. If we try to impose that (3) satisfy (5), this would require

which does not hold in any of the known solutions.

The closest one comes is when $W$ is 4-dimensional. The quadrilinear form, $f$, is just the 4-index $\epsilon$-symbol, and $\wedge^2 W$ is decomposable into self-dual and anti-self-dual forms $\wedge^2 W = \wedge^2_+ W \oplus \wedge^2_- W$ with respect to this form. The action of $\wedge^2_- W$ on $W$ is the one induced from (3), while the action of $\wedge^2_+ W$ on $W$ is the one induced from $\alpha\wedge\beta: X \mapsto X - [\alpha,\beta,X]$ These signs are made slightly more transparent by mapping $\mathbb{R}^4$ to the quaternions $\mathbf{X} = \frac{1}{2}\begin{pmatrix} X_4 + i X_3& i X_1 - X_2\\ i X_1 + X_2& X_4 - i X_3 \end{pmatrix}$ where we’ve represented the unit quaternions by $i$ times the Pauli matrices. In this notation, the inner product is $\tfrac{1}{2} (X, Y) = Tr(\mathbf{X}^\dagger \mathbf{Y})$ The matrix representing $[X,Y,Z]$ is $\begin{aligned} [X,Y,Z]\simeq & \tfrac{4}{3}\left( \mathbf{X} \mathbf{Y}^\dagger \mathbf{Z} +\mathbf{Y} \mathbf{Z}^\dagger \mathbf{X} +\mathbf{Z} \mathbf{X}^\dagger \mathbf{Y} \right.\\ &\left. - (\mathbf{Z} \mathbf{Y}^\dagger \mathbf{X} +\mathbf{Y} \mathbf{X}^\dagger \mathbf{Z} +\mathbf{X} \mathbf{Z}^\dagger \mathbf{Y} ) \right) \end{aligned}$ In this notation, $SO(4)\sim SU(2)_L\times SU(2)_R$ acts by conjugation $\mathbf{X} \mapsto g_L \mathbf{X} g_R^{-1}$ whereas (3) would have corresponded to $\mathbf{X} \mapsto g_L \mathbf{X} g_R$, which fails to satisfy the group law for $SU(2)_R$.

Anyway, introducing the covariant derivative^{2},
$D_\mu \mathbf{X} = \partial_\mu \mathbf{X} + A^L_\mu \mathbf{X} - \mathbf{X} A^R_\mu$
we can write the action as

where $k\in\mathbb{Z}$,

and

is the Chern-Simons action at level-1 for $SU(2)_L$ and level-($-1$) for $SU(2)_R$.

The supercharges are spacetime spinors in the $S$ of $Spin(8)$, and the supersymmetry variations

While it’s probably unsurprising, Bandres *et al* verify that the rest of the $OSp(8|4)$ superconformal algebra holds at the classical level, as well. Perhaps a good challenge for our ERGE friends would be to check that (7) is superconformal at the quantum level.

Parity is implemented, in this theory, in a slightly nonstandard way: accompanying a reflection in one of the spatial coordinates, is an exchange of the two $SU(2)$s and Hermitian conjugation on the matrices $\Phi$ and $\Psi$,
$\begin{gathered}
A^{L} \leftrightarrow A^R\\
\Phi^I \to \Phi^{I\dagger}\\
\Psi \to \gamma^1 \Psi^\dagger
\end{gathered}$
A Majorana mass term, $\psi\psi$ is a pseudo-scalar in 2+1 dimensions which, as noted by Bandres *et al* accounts the requisite sign in the transformation under parity of the second line of (8).

Are there other realizations, where we demand that (3), (4), (5) hold for just some subalgebra, $\mathfrak{g}\subset so(W)$? An obvious guess would be to replace $W=\mathbb{H}$ by $W=\mathbb{H}(n)$, the space of $n\times n$ *matrices* of quaternions. Bandres *et al* looked for other, more nontrivial, examples, but didn’t find any.

#### Update: Moduli Space

Van Raamsdonk points out that the moduli space — the space of zeroes of the scalar potential in (8), modulo $SO(4)$ gauge transformations — of the Bagger-Lambert model is $(\mathbb{R}^8\times \mathbb{R}^8)/SO(2)$. This is because a zero of the potential requires that all 8 of the $\phi^I$ lie in a common 2-plane in $W$. Configurations which differ by a rotation *within* that 2-plane are gauge-equivalent, so the moduli space seems to be $V\otimes\mathbb{R}^2/SO(2)$. Rotations in the *orthogonal* 2-plane comprise a residual unbroken $SO(2)$ gauge symmetry, which does does not act on the moduli space.

This looks like a puzzle, because it’s hard to see how the bosonic spectrum (15 massless scalars and a nondynamical $U(1)$ gauge field) could be $N=8$ supersymmetric. Fortunately, Mukhi and Papageorgakis ride to the rescue. They show that, when one integrates out the massive modes, the $U(1)$ gauge field actually becomes dynamical^{3}. A dynamical $U(1)$ gauge field can be dualized to a circle-valued scalar, so the full moduli space is
$\frac{\mathbb{R}^8\times \mathbb{R}^8}{SO(2)}\, \times\, S^1$
which is the desired answer for the moduli space of a pair of M2-branes. (If I were a little more energetic, I would attempt to show that this actually gets the discrete identifications right, and that the moduli space is $(\mathbb{R}^8\times \mathbb{R}^8)/\mathbb{Z}_2$.)

^{1} In 3+1 dimensions, the Lorentz group is $SL(2,\mathbb{C})$, and we need to distinguish between the two distinct two-dimensional representations. Spinors in the $\mathbf{2}$, $\psi_\alpha$, carry undotted indices. Their Hermitian conjugates, $\overline{\psi}_{\dot{\alpha}}$, carry dotted indices and transform in the $\overline{\mathbf{2}}$. In 2+1 dimensions, there’s only one type of spinor, and we can impose a Majorana condition.

^{2} If you want to insist on using the previous notation, we can write this as
$D_\mu X = \partial_\mu X + [A^L_\mu- A^R_\mu;X]$
where $[\omega;X]: \wedge^2W\otimes W\to W$ is the action of 2-forms on $W$, induced from (3), and we explicitly separate out the self-dual and anti-self-dual pieces.

^{3} Note that, from the underlying parity-invariance of the theory, this $U(1)$ must have vanishing Chern-Simons coefficient, and so is a *massless* dynamical gauge field.

## Re: Bagger-Lambert

Thanks for the very useful review.

The day before yesterday I had spent an hour trying to see whether (3), (4), (5) secretly encodes an $L_\infty$-algebra structure. So far I haven’t found a satisfactory solution.

The discussion in the appendix of J. Bagger, Comments on multiple M2-branes shows that (3)-(5) is equivalent to a vector space $A \oplus B$ equipped with a skew bracket which satisfies the Joacobi identity everywhere except on $\wedge^3 A$. There the Jacobi identity plain fails (not “up to something”).

That looks odd. If that is all there is, I’d say there might be little point in talking about “3-algebras” in this context. Instead, it would seem that we are just talking about nothing but the volume form on $\mathbb{R}^4$.

But then of course, the way this arises from gauge transformations with not one but two gauge parameters suggests that there is more going on after all. I’ll need to think about this more.