What is ABJM, and what can we calculate?

Can the chiral and vector multiplets we already know describe a strongly interacting theory with twelve real supercharges? They can: ABJM is named after Aharony, Bergman, Jafferis, and Maldacena, who constructed a family of three-dimensional supersymmetric Chern–Simons–matter theories. We will specify one member of that family precisely enough to count its fields, recognize its interactions, and distinguish it from supersymmetric Yang–Mills theory.

The preparation is one-flavor SQED, matrix-valued matter and traces, and the flat-space Chern–Simons coupling. No brane construction, large-rank limit, or curved-space supersymmetry is needed here.

First calculation: identify what the numbers count

Fix integers $N\geq1$ and $k>0$. The theory is

$$\boxed{\text{3d ABJM},\qquad G=U(N)_k\times U(N)_{-k},\qquad \text{four chiral multiplets }A_1,A_2,B_1,B_2.} \tag{AB01.1}$$

The subscripts are Chern–Simons levels, not dimensions of representations. The two $U(N)$ factors have equal size and opposite levels. Their matter content, written in a three-dimensional $\mathcal N=2$ language, is:

Multiplets	Representation under the first and second factors	Scalar entries
$A_1,A_2$	$(\boldsymbol N,\overline{\boldsymbol N})$	Two complex $N$-by-$N$ matrices
$B_1,B_2$	$(\overline{\boldsymbol N},\boldsymbol N)$	Two complex $N$-by-$N$ matrices with the reverse index direction
First vector multiplet	Adjoint of the first factor	Gauge connection $a_\mu$, scalar $\sigma$, auxiliary $D$
Second vector multiplet	Adjoint of the second factor	Gauge connection $\widehat a_\mu$, scalar $\widehat\sigma$, auxiliary $\widehat D$

Each chiral also contains a two-component complex fermion and a complex auxiliary. Each vector contains gauginos. A hat labels the second gauge factor; it is not complex conjugation. In particular, $B_i$ is a chiral multiplet, not the antichiral partner of $A_i$. This is the matter assignment in BK, section 2.1, also used in KWY, (4.2).

For $N=2$, each matter matrix has four complex entries. Thus there are sixteen complex scalar entries, or thirty-two real ones, and sixteen two-component complex matter-fermion fields. Each $\mathfrak u(2)$ algebra has four real generators: the two nodes together have eight generators and rank four. These are counts of fields before gauge constraints, equations of motion, or a choice of vacuum. They are not counts of gauge-invariant particles or states in the physical Hilbert space.

The other symbol, $\mathcal N$, counts supersymmetry. A minimal Lorentzian spinor in three dimensions has two real components. Consequently, generic ABJM has $\mathcal N=6$, meaning twelve real Poincaré supercharges, while this presentation makes an $\mathcal N=2$ subalgebra, or four real supercharges, manifest. Changing matrix size does not change that count. Superconformal partners are additional generators, counted separately. AB, introduction

At $k=1,2$, the standard compact theory has enhanced $\mathcal N=8$ supersymmetry. The additional currents involve monopole operators; they cannot be found by counting elementary fermions in the table. This is a field-theoretic result of BK, sections 2.8–2.10, beyond the original paper’s conjecture. Level one also has a free sector, so enhancement is not a claim that every sector is interacting. We will use the generic twelve-charge description unless an exceptional level is specified.

Specify the action, not only the field list

We work in flat space, with no masses or FI terms. Supercharge counts refer to Lorentzian signature; displayed component couplings use the site’s Euclidean continuation. All Euclidean fermions and their antichiral partners are independent integration variables.

Use Hermitian gauge matrices and the ordinary fundamental trace $\operatorname{Tr}_N(t^at^b)=\delta^{ab}/2$. The separately normalized trace $\operatorname{tr}_N=\operatorname{Tr}_N/N$ must not replace it in the action. Define $f=da-i\,a\wedge a$, and similarly $\widehat f$. A fundamental derivative is $\partial_\mu-ia_\mu$; the two connections act on opposite sides of a bifundamental matrix.

The defining ingredients are the gauge-covariant canonical chiral kinetic terms, the supersymmetric Chern–Simons terms at opposite levels, and a particular quartic superpotential. With spinor indices contracted as in the Chern–Simons lesson, define the level-free functional

$$\begin{aligned} \mathcal C[V]={}& i\int\operatorname{Tr}_N \left(a\wedge da-\frac{2i}{3}a\wedge a\wedge a\right)\\ &+\int d^3x\,\operatorname{Tr}_N (-2\sigma D+2i\lambda\widetilde\lambda). \end{aligned} \tag{AB01.2}$$

Here $V$ denotes the whole vector multiplet, and $\lambda\widetilde\lambda=\lambda^{\mathsf T}C\widetilde\lambda$ uses $C=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)$ on spinor indices; the trace contracts gauge indices. The second node has the same functional with hats. In terms of familiar supersymmetric building blocks,

$$S_E=\frac{k}{4\pi}\mathcal C[V] -\frac{k}{4\pi}\mathcal C[\widehat V] +S_{\rm kin}^{\rm cov}[A_i,B_i] +S_F[W,\widetilde W], \qquad W=\frac{4\pi}{k}\operatorname{Tr}_N (A_1B_1A_2B_2-A_1B_2A_2B_1). \tag{AB01.3}$$

$S_{\rm kin}^{\rm cov}$ is the canonical chiral kinetic action with both gauge factors included. $S_F$ integrates the top components of $W$ and its antichiral counterpart $\widetilde W$, as in the superpotential construction. On the Lorentzian physical slice the latter is the conjugate interaction. We fix the holomorphic field phases so the displayed coefficient agrees with AB v4, (2.5). A field rephasing would require translating that convention.

This formula specifies the interaction; its complete matrix expansion will require following the ordered gauge indices. The opposite-level trace pairing also appears in KWY, (4.3). It is not a positive kinetic metric on the two gauge algebras: neither term is a Yang–Mills kinetic energy.

The polynomial is quartic, but that fact alone does not establish a quantum conformal theory. ABJM has a particular relation between its superpotential coefficient and its levels. An arbitrary quartic coupling need not have the same extended supersymmetry or quantum scaling properties; see KWY, section 2, printed p. 7.

A dimensional check

In mass units, $[\partial_\mu]=1$ and a Lagrangian density in three dimensions has dimension three. A canonical scalar kinetic term gives $2+2[A_i]=3$, hence $[A_i]=[B_i]=1/2$. The fermion kinetic term gives $[\psi]=1$, while $[\widetilde F F]=3$ gives $[F]=3/2$. Gauge covariance and the Chern–Simons partners then give

$$[a]=[\sigma]=1,\qquad [D]=2,\qquad [\lambda]=\frac32,\qquad [k]=0. \tag{AB01.4}$$

The chiral measure has $[d^2\theta]=1$, so a superpotential must have dimension two. Four scalar factors supply precisely two; $4\pi/k$ is dimensionless. These are engineering dimensions of the defining fields. They do not independently prove exact operator dimensions at an interacting fixed point. Also, a gauge-charged elementary field is not itself a local gauge-invariant observable.

Why the vector multiplet is not a Yang–Mills multiplet

First turn off matter and linearize a gauge connection around zero on a contractible patch. For each Lie-algebra generator, varying its Chern–Simons action gives $\epsilon^{\mu\nu\rho}\partial_\nu a_\rho=0$, or $da=0$. Locally $a=d\alpha$, so the solution is gauge. The same argument applies to the other node, since $-k$ is also nonzero.

This statement includes Lorentzian wave solutions. For a Fourier amplitude $\varepsilon_\mu$ at any nonzero covector $p_\mu$, the equation is $p_\mu\varepsilon_\nu-p_\nu\varepsilon_\mu=0$. Choose a component $p_j\neq0$. Then $\varepsilon_\mu=p_\mu\varepsilon_j/p_j$, a pure-gauge amplitude, even when $p$ is null. There is no independent local photon polarization. Nontrivial topology can still leave holonomies and topological states, and matter sources change the equation. Absence of a Yang–Mills photon does not mean absence of gauge interactions.

The scalar $\sigma$ is constrained too. To see exactly how, select the usual bosonic physical slice for the matter scalars and define the Hermitian matrices

$$\mu_L=\sum_{i=1}^2(A_iA_i^\dagger-B_i^\dagger B_i), \qquad \mu_R=\sum_{i=1}^2(B_iB_i^\dagger-A_i^\dagger A_i).$$

Before choosing that slice, daggers here are replaced by independent antichiral scalars. The part linear in the vector auxiliaries follows from (AB01.2) and the charge-weighted matter coupling:

$$\mathcal L_D= \operatorname{Tr}_N\left[ D\left(\mu_L-\frac{k}{2\pi}\sigma\right) +\widehat D\left(\mu_R+\frac{k}{2\pi}\widehat\sigma\right) \right]. \tag{AB01.5}$$

There is no $D^2/g_{\rm YM}^2$ term. Variation with respect to each matrix component therefore imposes

$$\boxed{\sigma=\frac{2\pi}{k}\mu_L,\qquad \widehat\sigma=-\frac{2\pi}{k}\mu_R.} \tag{AB01.6}$$

These relations determine the vector scalars from matter. They do not set the matter density to zero and do not leave the independent Coulomb scalar of zero-level SQED.

For a concrete check, take $N=1$. Put $u=|A_1|^2+|A_2|^2-|B_1|^2-|B_2|^2$. Then $\mu_L=u$, $\mu_R=-u$, and both scalar solutions equal $2\pi u/k$. Changing the sign of the second level is essential for this equality. This is an auxiliary-field check, not a derivation of the global vacuum space.

For comparison, AB, (2.1) and its following auxiliary equation gives $\sigma_{\rm AB}=-(4\pi/k)\sum_a\operatorname{Tr}_N(t^a\mu_L)t^a$. Completeness of the full $\mathfrak u(N)$ basis gives $2\sum_a\operatorname{Tr}_N(t^a\mu_L)t^a=\mu_L$. Its equation therefore agrees with ours under $\sigma_{\rm here}=-\sigma_{\rm AB}$. This is a translation of the eliminated bosonic constraint, not a prescription for Wick rotating every source field or its auxiliary contour. Our Euclidean off-shell action is fixed by (AB01.2) and the inherited chiral action; the holomorphic coefficient agrees separately with (2.5).

Algebraic variation here is not an assertion that a real Euclidean $D$ integral converges. Without a quadratic regulator it imposes a constraint on an appropriate continued contour. A supersymmetric Yang–Mills regulator can supply a Gaussian definition; removing that regulator does not add propagating Yang–Mills fields to the defining theory.

Keep the compact gauge group

We use the ordinary product of compact groups

$$U(N)\times U(N),\qquad U(N)=\frac{SU(N)\times U(1)}{\mathbb Z_N}, \tag{AB01.7}$$

with no further quotient of the two factors. The internal $\mathbb Z_N$ identifies a common central transformation in the two descriptions of a single $U(N)$ matrix.

In a diagonal magnetic patch around a two-sphere, the allowed fluxes are $\int f/(2\pi)=\operatorname{diag}(n_1,\ldots,n_N)$ and an analogous hatted matrix, with integers in each factor, up to permutations. These specify admissible gauge bundles or disorder boundary data, before imposing the Gauss-law constraints on physical states. BK, (2)

Locally write $a=a_{\rm su}+a_0\boldsymbol1_N$, where $\operatorname{Tr}_N a_{\rm su}=0$. Substitution in the quadratic Chern–Simons form gives

$$\operatorname{Tr}_N(a\wedge da) =\operatorname{Tr}_N(a_{\rm su}\wedge da_{\rm su}) +N a_0\wedge da_0. \tag{AB01.8}$$

Thus the central levels are $Nk$ and $-Nk$ in this local decomposition. Globally the traceless and central pieces are linked by the quotient in (AB01.7); this is not a freely chosen product of an $SU(N)$ theory and an independent $U(1)$ theory. In particular, $\int da_0/(2\pi)=\sum_i n_i/N$ need not be an integer by itself.

The supersymmetric theory is a spin theory. The integer $k$ labels its fixed compact, gauge-invariant supersymmetric Chern–Simons definition with the fermion regulator understood. On nontrivial bundles the action is defined by its spin Chern–Simons functional, not by pretending that one global $a$ exists. Replacing $U(N)$ by $SU(N)$, changing the flux lattice, or silently changing to the normalized trace changes the theory.

What calculations does this definition open?

There are several different questions. The classical action determines auxiliary constraints and vacuum equations. Gauge-invariant operators require more than the elementary field table, particularly when magnetic flux is inserted. A supersymmetric curved background makes sphere partition functions and certain Wilson loops accessible to localization; KWY, section 4 is a concrete example. Localization evaluates specified observables with their measures and prescriptions, not arbitrary correlation functions.

For ordinary perturbation theory, large $k$ at fixed $N$ weakens the interactions; at large rank the relevant parameter is $N/k$. Neither a large matrix nor a supersymmetric action alone guarantees weak coupling. A sphere calculation also requires a specified rigid background and preserved supercharges. The flat-space label $\mathcal N=6$ does not say that every curved metric preserves twelve of them.

Continue with Chern–Simons gauge fields with non-Abelian matter to put both gauge connections into the matter derivative and check the opposite-level action without losing matrix order. The checkpoint here is to use given $N$ and $k$ to write the field representations and level pair, explain the auxiliary constraint, and state which supersymmetry count is being used.

Exercises

1. A rank is not a supersymmetry label

Take $U(3)_5\times U(3)_{-5}$. Count the real scalar entries, two-component complex matter-fermion fields, real gauge-algebra generators, and total gauge rank. State the generic full and manifest numbers of real Poincaré supercharges. Explain why the first count is not a count of physical scalar particles.

Hint 1

Count entries in one complex $3$-by-$3$ matrix before counting the four multiplets.

Hint 2

The dimension of $\mathfrak u(N)$ is $N^2$, while its rank is $N$. Gauge constraints act on field configurations and states.

Solution

There are $4\cdot9=36$ complex scalar entries, hence $72$ real scalar entries, and $36$ two-component complex fermion fields. The two algebras contain $2\cdot9=18$ real generators and have total rank $2\cdot3=6$. Their gauge connections therefore have $3\cdot18=54$ real component functions before gauge constraints.

At $k=5$ the full theory has twelve real Poincaré supercharges, four manifest in this organization. The number six appearing as the total gauge rank is unrelated to $\mathcal N=6$. Scalar entries transform under gauge redundancy; the physical Hilbert space obeys Gauss law, and a particle interpretation also depends on the vacuum and dynamics. One cannot identify the $72$ real entries with $72$ physical scalar particles.

2. Diagnose a proposed modification

A proposed action replaces $4\pi/k$ in (AB01.3) by a parameter $h$ and adds $\operatorname{Tr}_N f_{\mu\nu}f^{\mu\nu}/(4g^2)$. Determine the engineering dimensions of $h$ and $g^2$. Does dimensional analysis prove that this is the same conformal theory? For the undeformed pure Chern–Simons theory, also test the Lorentzian null covector $p_\mu=(1,1,0)$ directly for a physical photon amplitude.

Hint 1

The superpotential density has dimension two; $[f]=2$.

Hint 2

Use the antisymmetric equation $p_\mu\varepsilon_\nu-p_\nu\varepsilon_\mu=0$ without dividing by $p^2$.

Solution

The quartic has dimension two, so $[h]=0$. Since $[f^2]=4$, $[g^2]=1$. The Maxwell term introduces a scale and propagating vector modes when its supersymmetric completion is retained. A dimensionless $h$ is only classically marginal; a general value is not the ABJM relation and does not by itself establish extended supersymmetry or quantum conformality. An infrared construction might remove a regulator and tune the required couplings, but equality at finite $g^2$ has not been shown.

For the null covector, the $(0,1)$ equation gives $\varepsilon_1=\varepsilon_0$ and the $(0,2)$ equation gives $\varepsilon_2=0$. Thus $\varepsilon_\mu=\varepsilon_0p_\mu$, exactly a gauge amplitude. The argument does not fail at a null momentum. It concerns a local mode of the pure Chern–Simons action, not global holonomies.

3. Check the center without changing the group

For one $U(3)_2$ factor, take diagonal flux $(n_1,n_2,n_3)=(1,0,0)$. Find the flux of $a_0=\operatorname{Tr}_3a/3$ and the traceless flux. Is this a permitted $U(3)$ configuration? Compare it with a purely central configuration having unit $a_0$ flux. Determine the coefficient of the central Chern–Simons term, and explain the error in replacing the ordinary trace by $\operatorname{Tr}_3/3$ at fixed declared level.

Hint 1

Subtract one third of the trace from each diagonal entry.

Hint 2

A standalone central connection is proportional to the identity, so all three eigenvalue fluxes agree. Use $\operatorname{Tr}_3\boldsymbol1_3=3$.

Solution

The central flux is $1/3$, and the traceless part is $\operatorname{diag}(2/3,-1/3,-1/3)$. Their combination is the integral $U(3)$ flux $(1,0,0)$, so it is allowed. The two pieces should not separately be treated as globally independent bundles. A purely central unit flux instead gives eigenvalue flux $(1,1,1)$ and vanishing traceless flux.

Equation (AB01.8) gives central level $Nk=6$, or Euclidean coefficient $6i/(4\pi)$ multiplying $a_0\wedge da_0$. Dividing the trace by three without translating the level would give central level two and rescale the non-Abelian pairing too. It is not a notation change. For example, on a spin four-manifold with an embedded unit-flux line bundle satisfying $\int c_1^2=2$, the ordinary level-two extension changes $S_E$ by $4\pi i$, leaving $e^{-S_E}$ unchanged. The wrongly divided expression changes it by $4\pi i/3$. Thus the altered expression need not even define the originally declared compact theory.