Where does N=6 supersymmetry come from?

Before this lesson: From auxiliary fields to a sextic scalar potential; Fixed points, scaling, and conformal symmetry.

Four chiral multiplets make four real supercharges visible. Why does ABJM have twelve? A larger symmetry of the scalar potential would be suggestive, but a supersymmetry transformation also changes fermions and gauge fields. We need symmetries of the whole action, and then an argument that they generate additional supercharges.

Start with the complete scalar potential and the distinction between engineering dimensions and conformal symmetry. We work in flat three-dimensional spacetime with compact $U(N)_k\times U(N)_{-k}$ , $N\geq1$ , positive integer $k$ , canonical matter normalization, ordinary fundamental traces, and no mass or FI deformation. Each gauge factor has rank $N$ . The global group has no additional common-center quotient.

The generic theory has $\mathcal N=6$ : twelve real Lorentzian Poincaré supercharges. The chosen $\mathcal N=2$ superspace displays four. A conformal theory has additional conformal supercharges, which are not included in this twelve. We will establish the generic enhancement argument; extra symmetries at exceptional levels require a separate analysis of monopole sectors.

Scalars and fermions do not form the same four

The independent chiral scalars are $A_1,A_2:V_R\to V_L$ and $B_1,B_2:V_L\to V_R$ . On the physical scalar slice, assemble

Y^I=(A_1,A_2,B_1^\dagger,B_2^\dagger),\qquad I=1,\ldots,4. \tag{AB06.1}

All four now map $V_R$ to $V_L$ . Thus a constant unitary matrix $U^I{}_J$ may mix them while commuting with their gauge transformation: $Y^I\mapsto G_LY^IG_R^{-1}$ . The dagger is essential: $B_1$ itself has the opposite color representation. This packaging of scalar components does not make $B_1^\dagger$ a chiral superfield.

The full R-symmetry acts as $Y^I\mapsto U^I{}_JY^J$ with $U\in SU(4)$ . To identify its manifest R-generator, use

R=\frac12\operatorname{diag}(1,1,-1,-1). \tag{AB06.2}

Every original chiral scalar has charge $1/2$ , whereas the conjugated $B$ scalars have charge $-1/2$ . A chiral fermion has one unit less R-charge than its scalar. Consequently the four fermions with color direction $V_R\to V_L$ have charges $(-1/2,-1/2,+1/2,+1/2)$ . They organize as a conjugate four, denoted $\psi_I$ , rather than as another copy of $Y^I$ .

Here is an explicit way to see the subgroup labels. Let $\psi_{A_a}$ and $\psi_{B_{\dot a}}$ be the two-component chiral fermions. Charge conjugation, denoted by $c$ , turns a Lorentzian $B$ fermion into one with the opposite color representation. With $\epsilon_{12}=1$ , choose the flavor bases $\psi_a=\epsilon_{ab}\psi_{A_b}$ and $\psi_{2+\dot a}=\epsilon_{\dot a\dot b}\psi_{B_{\dot b}}^c$ . The identity $\epsilon U=U^*\epsilon$ for $SU(2)$ converts the natural doublet into its conjugate. These are basis choices for the fermions; spinor charge conjugation is not an extra independent chiral field.

Fields after eliminating algebraic variables	Color representation	Full R-representation	Engineering dimension
$Y^I$	$(N,\overline N)$	$\mathbf4$	$1/2$
$\psi_I$	$(N,\overline N)$	$\overline{\mathbf4}$	$1$
Their physical conjugates	$(\overline N,N)$	Conjugate representations	Same
$a_\mu,\widehat a_\mu$	The two adjoints	Singlets	$1$

The vector scalars, gauginos, and auxiliary fields used in the superspace construction are algebraic variables here, not extra propagating matter. Once eliminated, their effects remain in the scalar and fermion interactions. In Euclidean calculations one may complexify the fields and transformation parameters; the Lorentzian real count should not then be doubled by counting each complexified variable as an additional physical supercharge.

Two known symmetries generate a larger one

The needed input from AB v4, sections 2.1–2.2 is the supersymmetric construction of the full $\mathcal N=3$ action, including its fermion couplings. The opposite Chern–Simons levels and algebraic adjoint superfields fix

W=2t\,\operatorname{Tr} (A_1B_1A_2B_2-A_1B_2A_2B_1),\qquad t=\frac{2\pi}{k}. \tag{AB06.3}

The previous superpotential lesson derived this coefficient by eliminating those adjoints. Its $\mathcal N=3$ interpretation supplies an $SU(2)_R$ rotating $(A_i,B_i^\dagger)$ simultaneously for $i=1,2$ . The existence of this symmetry of the complete supersymmetric action is an input, not a conclusion drawn from the bosonic calculation below.

There is also an independently visible flavor symmetry $SU(2)_A\times SU(2)_B$ . Indeed, $W=t\epsilon^{ab}\epsilon^{\dot a\dot b} \operatorname{Tr}(A_aB_{\dot a}A_bB_{\dot b})$ . Both epsilon tensors are invariant, and every kinetic or vector coupling sums over the corresponding flavor index. This argument acts on entire chiral multiplets, including their fermions. For example the superpotential Yukawa coupling contains its matrix of second derivatives,

\mathcal L_{W,\mathrm{fermion}} \ \propto\ W_{mn}(\phi)\,\psi_m\psi_n+\text{conjugate}. \tag{AB06.4}

Here $m,n$ label chiral species and color entries, and the spinor contraction is antisymmetric. For a linear flavor transformation $T$ , invariance of $W(T\phi)$ implies $T^{\mathsf T}W''(T\phi)T=W''(\phi)$ . It therefore preserves the Yukawa term too. The two fermions anticommute, making their contracted bilinear symmetric in $m,n$ , as required by $W_{mn}=W_{nm}$ . Omitting the Yukawa term would not define the same supersymmetric theory.

We can now calculate the symmetry generated by these groups. Let $E_{IJ}$ be the $4\times4$ matrix with one nonzero entry in row $I$ , column $J$ . For finding the Lie algebra, allow complex linear combinations of generators; the compact real form is recovered by taking anti-Hermitian matrices. The flavor generators include $E_{12},E_{21},E_{34},E_{43}$ . The original R-symmetry supplies

J_+=E_{13}+E_{24},\quad J_-=E_{31}+E_{42}, \quad [J_+,J_-]=2R.

Their commutators include

[E_{12},J_+]=E_{14},\qquad [E_{21},J_+]=E_{23},\qquad [E_{43},E_{14}]=-E_{13}. \tag{AB06.5}

Thus $E_{24}$ is available separately too. Conjugate generators supply the four reverse off-block directions. Together there are four off-diagonal directions inside the two blocks, eight between them, and three independent traceless diagonal directions: fifteen in total. This is $\mathfrak{su}(4)$ in its compact real form.

Because each starting transformation preserves the full action, their commutators do too. Moreover $SU(2)_R$ already rotates supercharges, so the resulting simple Lie algebra cannot act only as flavor symmetry commuting with every supercharge. This is the crucial distinction from merely discovering an accidental scalar-potential symmetry.

Why the supercharges form a real six

The supersymmetry directions can be written as an antisymmetric tensor $\omega^{IJ}_\alpha$ , where $\alpha=1,2$ is a Lorentzian Majorana spinor-component index. With $\epsilon_{1234}=1$ , impose

(\omega^{IJ}_\alpha)^* =\frac12\epsilon_{IJKL}\omega^{KL}_\alpha. \tag{AB06.6}

This reality condition is preserved by $SU(4)$ . For each $\alpha$ , write $p=\omega^{12}$ , $q=\omega^{13}$ , $r=\omega^{14}$ . The other independent entries are $\omega^{34}=p^*$ , $\omega^{24}=-q^*$ , $\omega^{23}=r^*$ . Three complex numbers subject to this completion give six real directions, not six complex ones. Including the two real spinor components gives twelve real Poincaré supercharges.

Under the manifest subgroup the six splits as

\mathbf6=(\mathbf1,\mathbf1)_{+1} \oplus(\mathbf1,\mathbf1)_{-1} \oplus(\mathbf2,\mathbf2)_0. \tag{AB06.7}

The subscripts are charges under (AB06.2). The two singlets, related by reality, describe the manifest $\mathcal N=2$ directions; the four real mixed directions supply the rest. In tensor language the scalar variation has the allowed contraction $\omega^{IJ}\psi_J$ : a six times a conjugate four maps to a four. This identifies the representation, not the complete interacting transformation rule.

There is also a direct way to understand why transformed supercharges remain symmetries. If $Q$ and a bosonic generator $T$ preserve the action, then $[T,Q]$ preserves it. Repeated commutators generate the six above. AB’s full-action construction implements this argument; the superconformal R-algebra is locally $\mathfrak{so}(6)\simeq\mathfrak{su}(4)$ . Using $SU(4)$ on elementary fields is convenient because its four is a spinor representation of $SO(6)$ . This local statement does not determine all global quotients involving gauge centers and line operators.

This constructs the generic six supersymmetry directions; it does not exclude extra ones in exceptional theories or sectors. The result concerns the theory in its conformal vacuum. Counting dimension-three interactions alone would not prove quantum conformality; the superconformal construction and its symmetry protection are the physical input here. Nor does a generic curved background automatically preserve all twelve charges.

Watch two interactions exchange strength

The scalar potential supplies a stringent independent check. Set $L=\sum_IY^IY_I^\dagger$ and $R_c=\sum_IY_I^\dagger Y^I$ ; the subscript distinguishes the right-node density from the R-generator. Expanding the positive squares of the previous lesson gives

\begin{aligned} \frac{V}{t^2}={}&-\frac13\operatorname{Tr}(L^3) -\frac13\operatorname{Tr}(R_c^3) +2\sum_J\operatorname{Tr}(LY^JR_cY_J^\dagger)\\ &-\frac43\sum_{I,J,K}\operatorname{Tr} (Y^IY_K^\dagger Y^JY_I^\dagger Y^KY_J^\dagger). \end{aligned} \tag{AB06.8}

These are AB’s four contractions in (A.7), in the same scalar normalization. Every repeated flavor index contracts a fundamental with its conjugate; inserting $U$ and $U^\dagger$ cancels them in pairs. Color matrices retain their written order. The negative coefficients do not contradict positivity: (AB06.8) is the expanded form of the positive squares.

For a visible noncommuting example, take $N=2$ and real $v,w>0$ : $A_1=vE_{11}$ , $A_2=wE_{12}$ , $B_1=B_2=0$ . Initially $V_F=0$ and $V_{\rm CS}=t^2v^2w^2(v^2+w^2)$ . Now rotate components two and three of $Y$ :

\binom{Y^{2\prime}}{Y^{3\prime}} =\begin{pmatrix}c&s\\-s&c\end{pmatrix} \binom{Y^2}{Y^3},\qquad c=\cos\theta,\quad s=\sin\theta.

Thus $A_2'=cwE_{12}$ and $B_1'=-swE_{21}$ . This is an $SU(4)$ rotation of components, not a holomorphic change of the four original chiral superfields.

The only nonzero superpotential gradient is the one with respect to $B_2$ :

\frac{\partial W}{\partial B_2} =2t(A_1B_1'A_2'-A_2'B_1'A_1) =2tcs\,vw^2E_{11}.

Recomputing both density matrices in the vector terms yields

\begin{aligned} V_F'&=4t^2c^2s^2v^2w^4,\\ V_{\rm CS}'&=t^2v^2w^2 \left[v^2+(c^2-s^2)^2w^2\right]. \end{aligned} \tag{AB06.9}

Since $(c^2-s^2)^2+4c^2s^2=1$ , the total is unchanged. At $\theta=\pi/4$ , the two contributions become $t^2v^2w^4$ and $t^2v^4w^2$ . Neither contribution is separately invariant. All terms have dimension three because $[v]=[w]=1/2$ and $[t]=0$ .

Exercises

1. Change the superpotential coefficient

Replace $W$ by $\eta W$ , with constant complex $\eta$ , keeping the matter normalization and both Chern–Simons levels fixed. Use the preceding finite rotation to test the claimed full R-symmetry. What does this test say about the phase of $\eta$ ?

Hint 1

Eliminating the chiral auxiliaries multiplies $V_F$ by $|\eta|^2$ . The vector contribution is unchanged.

Hint 2

The starting configuration has $V_F=0$ . Compare its total potential with the rotated configuration for $cs\,vw\ne0$ .

Solution

The change in total potential is

V_\eta(Y')-V_\eta(Y) =4t^2c^2s^2v^2w^4(|\eta|^2-1).

For $|\eta|\ne1$ , the displayed $SU(4)$ rotation is not a symmetry even of the scalar action. Thus gauge invariance, R-charge two, and classical marginality of $W$ did not fix the enhanced theory.

This test only fixes the magnitude. A constant phase can be moved into a rephasing of one whole chiral multiplet, for example $A_1$ , which leaves its kinetic and vector couplings unchanged. The fermionic transformations must be expressed in the correspondingly rephased basis. The vanishing defect at $|\eta|=1$ is a necessary check, not by itself a proof of full supersymmetry.

2. Count the six without complex double counting

For one spinor component, write the full antisymmetric matrix $\Omega=(\omega^{IJ})$ in terms of $p,q,r$ . Show that $\Omega\Omega^\dagger=(|p|^2+|q|^2+|r|^2)\mathbf1_4$ . Explain what happens if you omit (AB06.6).

Hint 1

Reality fixes entries $34,24,23$ from $12,13,14$ , with a minus sign for $24$ .

Hint 2

The off-diagonal entries of $\Omega\Omega^\dagger$ cancel pairwise. For instance its $12$ entry is $qr-rq$ .

Solution

The matrix is

\Omega= \begin{pmatrix} 0&p&q&r\\ -p&0&r^*&-q^*\\ -q&-r^*&0&p^*\\ -r&q^*&-p^*&0 \end{pmatrix}.

Each row has squared norm $|p|^2+|q|^2+|r|^2$ ; distinct rows have zero Hermitian inner product. This proves the identity. These are commuting coefficients specifying a basis of symmetry directions, not Grassmann-odd transformation parameters being multiplied as ordinary numbers.

There are six real coefficients per Majorana spinor component, hence twelve physical real parameters in three-dimensional Lorentzian signature. Removing the condition produces the complexification of the six: useful for complexified equations, but it does not produce a new Lorentzian theory with twice the supersymmetry.

3. A bosonic symmetry can survive after supersymmetry is lost

Add the deformation $\Delta V=m_0^2\operatorname{Tr}(Y_I^\dagger Y^I)$ , with real $m_0^2>0$ , and change no fermion, gauge, or supersymmetry-transformation term. Does its $SU(4)$ invariance establish that the deformed action remains supersymmetric?

Hint 1

The scalar mass contracts a fundamental and its conjugate. Test its variation under one of the original scalar-to-fermion transformations.

Hint 2

Evaluate the variation on constant scalar and fermion profiles, with the conjugate fermion set to zero in the complexified algebra. A term proportional to $m_0^2Y^\dagger\psi$ cannot be removed by a spacetime total derivative on such profiles.

Solution

The mass term is $SU(4)$ invariant, but

\delta\Delta V=m_0^2\operatorname{Tr} \left[(\delta Y_I^\dagger)Y^I+ Y_I^\dagger\delta Y^I\right]

contains a nonzero scalar–fermion bilinear. For example choose a transformation with $\delta Y^1$ proportional to $\psi_2$ , and nonzero $Y_1^\dagger,\psi_2$ . The original action’s variation still cancels among its original terms; no new term cancels this added variation. Therefore the displayed deformation does not preserve the original supercharges. A supersymmetric mass deformation would require additional terms and modified transformations and must be checked as a new complete construction.

This counterexample explains why scalar-potential invariance was only a check in the enhancement argument. Its decisive inputs were the original supersymmetric action and the symmetries acting on all its fields.

You can now identify the four scalar components, the conjugate fermion representation, and the real six of supersymmetry directions, and explain why the tuned action is needed. The one-brane vacuum calculation combines zero scalar energy with compact gauge transformations and flux data. An R-symmetry calculation alone does not determine that quotient.

For a curved-space continuation, construct ABJM on the round three-sphere. Its chosen rigid subalgebra and curvature couplings must be specified in addition to the flat-space supersymmetry just established.

The original ABJM paper guide locates this calculation in the source argument and gives a changed example to check while reading.

The superconformal-bootstrap introduction teaches crossing and block positivity using a fully specified four-dimensional example. Its method is useful preparation for three-dimensional questions; its numerical coefficients belong to the stated four-dimensional theory.