Musings

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 spinors¹. 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 derivative², $$D_\mu \mathbf{X} = \partial_\mu \mathbf{X} + A^L_\mu \mathbf{X} - \mathbf{X} A^R_\mu$$ 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 dynamical³. 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 $(\mathbb{R}^8\times \mathbb{R}^8)/\mathbb{Z}_2$.)