Supersymmetry Cheatsheet

This appendix collects the supersymmetry conventions used in the course. It is not a replacement for a full course on supersymmetry; it is a working reference for the points that repeatedly enter modern CFT and AdS/CFT:

$$\text{supercharges} \quad\longrightarrow\quad \text{supermultiplets} \quad\longrightarrow\quad \text{shortening} \quad\longrightarrow\quad \text{protected CFT data} \quad\longrightarrow\quad \text{protected bulk fields}.$$

The guiding idea is simple. Ordinary conformal symmetry relates a primary operator to its descendants under $P_\mu$. Supersymmetry adds fermionic generators $Q$ that move between bosons and fermions. Superconformal symmetry adds fermionic lowering operators $S$, so that shortening and BPS bounds can be understood by exactly the same radial-quantization logic as ordinary unitarity bounds.

1. Conventions

Unless stated otherwise, this appendix uses mostly-Lorentzian notation in four dimensions. The conformal boundary of $\mathrm{AdS}_5$ has Lorentz group

$$SO(3,1) \simeq SL(2,\mathbb C),$$

so a Lorentz vector can be written as a bispinor,

$$x_{\alpha\dot\alpha}=x_\mu \sigma^\mu_{\alpha\dot\alpha}.$$

Undotted indices $\alpha,\beta=1,2$ transform in the $(\frac12,0)$ representation of $SL(2,\mathbb C)$, while dotted indices $\dot\alpha,\dot\beta=1,2$ transform in the $(0,\frac12)$ representation. We use antisymmetric tensors to raise and lower indices,

$$\psi^\alpha=\epsilon^{\alpha\beta}\psi_\beta, \qquad \psi_\alpha=\epsilon_{\alpha\beta}\psi^\beta, \qquad \epsilon^{12}=-\epsilon_{12}=1.$$

The complex conjugate of a left-handed Weyl spinor is right-handed:

$$(Q_\alpha)^\dagger=\bar Q_{\dot\alpha}.$$

In Euclidean signature, dotted and undotted spinors are independent complex spinors. This is why many supersymmetric localization and instanton computations look slightly different from Lorentzian unitarity arguments.

2. Poincare supersymmetry

The minimal structural input is that the supercharges square to translations. In four-dimensional $\mathcal N$-extended supersymmetry, the basic algebra without central charges is

$$\{Q^I_\alpha,\bar Q_{J\dot\alpha}\} = 2\delta^I{}_J\sigma^\mu_{\alpha\dot\alpha}P_\mu,$$ $$\{Q^I_\alpha,Q^J_\beta\}=0, \qquad \{\bar Q_{I\dot\alpha},\bar Q_{J\dot\beta}\}=0.$$

Here $I,J=1,\ldots,\mathcal N$ are $R$-symmetry indices. The supercharges carry engineering dimension

$$[Q]=\frac12,$$

because $Q^2\sim P$ and $[P]=1$.

With central charges included, the schematic form is

$$\{Q^I_\alpha,Q^J_\beta\} = \epsilon_{\alpha\beta}Z^{IJ},$$

where $Z^{IJ}=-Z^{JI}$ commutes with the Poincare generators. Central charges are crucial in massive BPS particle spectra, monopoles, dyons, and extended supersymmetric gauge theory. For local operator classification in a conformal theory, the superconformal algebra is usually more important than central extensions.

3. Multiplets

A supersymmetry multiplet is a representation generated by acting with supercharges on a state or operator. In CFT language, the best object is a superconformal primary $\mathcal O$:

$$K_\mu\mathcal O=0, \qquad S\mathcal O=0.$$

The ordinary conformal descendants are generated by $P_\mu$, while the supersymmetric descendants are generated by $Q$:

$$\mathcal O, \qquad Q\mathcal O, \qquad Q^2\mathcal O, \qquad \ldots, \qquad P_\mu\mathcal O, \qquad P_\mu Q\mathcal O, \qquad \ldots.$$

Since $Q$ is fermionic, each independent component can act at most once. A generic long multiplet therefore has many components. A shortened multiplet has fewer components because some $Q$‘s annihilate the primary or create null descendants.

The basic slogan is

$$\text{long multiplet} = \text{no extra annihilation by }Q,$$ $$\text{short multiplet} = \text{some }Q\text{'s annihilate the primary}.$$

Short multiplets are often protected: their dimensions and some correlation data cannot vary continuously, because a continuous change would require recombination with other multiplets.

4. Superconformal algebra

The ordinary conformal algebra in $d$ dimensions contains

$$P_\mu,\quad K_\mu,\quad D,\quad M_{\mu\nu}.$$

A superconformal algebra adds Poincare supercharges $Q$, special superconformal charges $S$, and an $R$-symmetry algebra $\mathfrak r$:

$$P_\mu,\quad K_\mu,\quad D,\quad M_{\mu\nu},\quad R,\quad Q,\quad S.$$

The most useful schematic relations are

$$[D,Q]=\frac{i}{2}Q, \qquad [D,S]=-\frac{i}{2}S,$$ $$[K,Q]\sim S, \qquad [P,S]\sim Q,$$ $$\{Q,\bar Q\}\sim P, \qquad \{S,\bar S\}\sim K,$$ $$\{S,Q\}\sim D+M+R.$$

The last line is the heart of BPS bounds. In radial quantization, $S$ is essentially the adjoint of $Q$. Thus positivity of a norm such as

$$\|Q|\mathcal O\rangle\|^2 = \langle \mathcal O|S Q|\mathcal O\rangle$$

implies inequalities relating scaling dimension, spin, and $R$-charges. Saturation of such an inequality means

$$Q|\mathcal O\rangle=0$$

for some component of $Q$, hence shortening.

5. Common superconformal algebras

The following list is only a practical map for AdS/CFT. Precise real forms and exceptional cases matter in detailed representation theory.

Dimension	Bosonic conformal algebra	Typical superconformal algebras	Common holographic role
$d=2$	$\mathfrak{sl}(2,\mathbb R)\oplus\mathfrak{sl}(2,\mathbb R)$ globally	$\mathcal N=(p,q)$ superconformal algebras	strings, $\mathrm{AdS}_3/\mathrm{CFT}_2$
$d=3$	$\mathfrak{so}(3,2)$	$\mathfrak{osp}(\mathcal N\vert 4)$	M2-branes, ABJM-like theories
$d=4$	$\mathfrak{so}(4,2)\simeq\mathfrak{su}(2,2)$	$\mathfrak{su}(2,2\vert\mathcal N)$, especially $\mathfrak{psu}(2,2\vert4)$	$\mathrm{AdS}_5/\mathrm{CFT}_4$
$d=5$	$\mathfrak{so}(5,2)$	exceptional superconformal algebras	5D SCFTs
$d=6$	$\mathfrak{so}(6,2)$	$\mathfrak{osp}(8^\ast\vert 2\mathcal N)$	M5-branes, $(2,0)$ theories

The canonical AdS/CFT example is

$$\mathcal N=4\ \mathrm{SYM} \quad\longleftrightarrow\quad \text{type IIB string theory on }\mathrm{AdS}_5\times S^5,$$

whose superconformal group is

$$PSU(2,2\vert4).$$

Its bosonic subgroup is

$$SO(4,2)\times SO(6)_R.$$

The first factor is the conformal group of the boundary CFT; the second is the isometry of $S^5$.

6. Four-dimensional $\mathcal N=1$ reference

Many superconformal ideas are easiest to remember in $\mathcal N=1$ language.

The $R$-symmetry is $U(1)_R$. A chiral primary satisfies

$$\bar Q_{\dot\alpha}\mathcal O=0.$$

At a four-dimensional $\mathcal N=1$ superconformal fixed point, a scalar chiral primary obeys

$$\Delta=\frac32 R.$$

This equation is not a classical engineering statement; it is a representation-theoretic shortening condition. It is why exact $R$-symmetries are so powerful in $\mathcal N=1$ SCFTs.

The superspace coordinates are

$$x^\mu,\qquad \theta^\alpha,\qquad \bar\theta^{\dot\alpha}.$$

A chiral superfield $\Phi$ satisfies

$$\bar D_{\dot\alpha}\Phi=0.$$

A superpotential deformation has the form

$$\delta S = \int d^4x\,d^2\theta\, W(\Phi)+\text{c.c.}$$

Since the action is dimensionless and $[d^2\theta]=1$, a superpotential term must have

$$\Delta(W)=3$$

at a conformal point. This is the supersymmetric version of marginality.

7. Four-dimensional $\mathcal N=2$ reference

The $R$-symmetry of a four-dimensional $\mathcal N=2$ SCFT is

$$SU(2)_R\times U(1)_r.$$

Useful protected operator sectors include:

Sector	Representative operator	Why it matters
Coulomb-branch chiral operators	$\mathcal E_r$	control Coulomb-branch geometry
Higgs-branch moment maps	$\mu^a$	encode flavor symmetry and current multiplets
Stress-tensor multiplet	$\mathcal J$	contains $T_{\mu\nu}$ and $SU(2)_R\times U(1)_r$ currents

In $\mathcal N=2$ theories, exactly marginal deformations sit in protected multiplets and are often related to complexified gauge couplings. This makes $\mathcal N=2$ a useful bridge between generic SCFTs and the maximally supersymmetric $\mathcal N=4$ theory.

8. Four-dimensional $\mathcal N=4$ SYM field content

The $\mathcal N=4$ vector multiplet contains:

Field	Lorentz representation	$SU(4)_R$ representation	Comment
$A_\mu$	vector	$\mathbf 1$	gauge field
$\lambda^A_\alpha$	left Weyl spinor	$\mathbf 4$	gauginos
$\bar\lambda_{A\dot\alpha}$	right Weyl spinor	$\bar{\mathbf 4}$	conjugate gauginos
$\Phi^{AB}=-\Phi^{BA}$	scalar	$\mathbf 6$	six real scalars

The antisymmetric scalar $\Phi^{AB}$ is equivalent to six real scalars $\phi^I$,

$$I=1,\ldots,6, \qquad \phi^I\in \mathbf 6\text{ of }SO(6)_R.$$

The two descriptions are related by $SO(6)$ gamma matrices,

$$\Phi^{AB} = \frac{1}{\sqrt2}(\Gamma_I)^{AB}\phi^I,$$

with a suitable reality condition. In many holographic formulas, the $SO(6)$ vector notation $\phi^I$ is cleaner; in representation theory, the $SU(4)_R$ notation is often cleaner.

The complexified gauge coupling is

$$\tau = \frac{\theta_{\rm YM}}{2\pi} + \frac{4\pi i}{g_{\rm YM}^2},$$

and the planar ‘t Hooft coupling is

$$\lambda=g_{\rm YM}^2 N.$$

In the standard $\mathrm{AdS}_5\times S^5$ normalization,

$$\frac{L^4}{\ell_s^4}\sim \lambda, \qquad g_s\sim \frac{g_{\rm YM}^2}{4\pi},$$

with precise numerical factors depending on convention.

9. $SU(4)_R\simeq SO(6)_R$ dictionary

Irreducible representations of $SU(4)$ are often labeled by Dynkin labels

$$[a,b,c].$$

Some essential examples are:

Representation	Dynkin label	Dimension	Physics
fundamental	$[1,0,0]$	$4$	left gauginos $\lambda^A$
anti-fundamental	$[0,0,1]$	$\bar 4$	right gauginos $\bar\lambda_A$
vector of $SO(6)$	$[0,1,0]$	$6$	scalars $\phi^I$
stress-tensor multiplet primary	$[0,2,0]$	$20'$	half-BPS scalar primary
half-BPS tower	$[0,p,0]$	variable	KK scalar tower on $S^5$
singlet	$[0,0,0]$	$1$	Konishi-like operators

The half-BPS single-trace scalar primaries are schematically

$$\mathcal O_p^{I_1\cdots I_p} = \operatorname{Tr} \left( \phi^{(I_1}\cdots \phi^{I_p)} \right) -\text{traces},$$

or, equivalently, using a null $SO(6)$ polarization $Y^I$ satisfying $Y\cdot Y=0$,

$$\mathcal O_p(x,Y) = Y_{I_1}\cdots Y_{I_p} \operatorname{Tr} \left( \phi^{I_1}(x)\cdots\phi^{I_p}(x) \right).$$

They transform in

$$[0,p,0]$$

and obey the protected dimension formula

$$\Delta=p.$$

For $p=2$, this is the bottom component of the stress-tensor multiplet. For $p\geq 2$, the corresponding supergravity modes are Kaluza—Klein harmonics on $S^5$.

10. Protected versus unprotected operators in $\mathcal N=4$ SYM

A quick classification useful for holography is:

Operator type	Example	Dimension
half-BPS single trace	$\operatorname{Tr}(Y\cdot\phi)^p$	protected, $\Delta=p$
stress-tensor multiplet	$\operatorname{Tr}(\phi^{(I}\phi^{J)})-\text{traces}$	protected, $\Delta=2$
conserved currents	$T_{\mu\nu}$, $J_\mu^A$	protected by conservation
Konishi operator	$\operatorname{Tr}(\phi^I\phi^I)$	unprotected
double-trace BPS composite	$:\mathcal O_p\mathcal O_q:$	receives $1/N$ mixing effects
long single-trace operators	spin-chain states	generally unprotected

The Konishi operator is often the first example of a long multiplet. At weak coupling it looks like a simple scalar bilinear, but its dimension is not fixed by symmetry. At strong coupling it becomes a stringy state rather than a light supergravity field.

This contrast is one of the most important lessons for AdS/CFT:

$$\text{protected short multiplets} \quad\leftrightarrow\quad \text{controlled supergravity/Kaluza--Klein modes},$$ $$\text{long unprotected multiplets} \quad\leftrightarrow\quad \text{massive string states or interacting bulk states}.$$

11. BPS shortening in practice

The full $\mathcal N=4$ shortening classification is elaborate, but the most frequently used case in AdS/CFT is the half-BPS tower

$$[0,p,0], \qquad \Delta=p.$$

These operators are annihilated by half of the Poincare supercharges. Because the number of annihilating supercharges is maximal among nontrivial local scalar operators, they are called half-BPS.

The protected dimension $\Delta=p$ is not a perturbative accident. It follows from the representation theory of $\mathfrak{psu}(2,2\vert4)$. The operator cannot continuously acquire an anomalous dimension without leaving the shortened representation. But leaving the shortened representation would require recombination with other multiplets carrying compatible quantum numbers. For isolated half-BPS multiplets, this recombination is forbidden.

The bulk scalar mass follows from the AdS scalar relation

$$m^2L^2=\Delta(\Delta-4)$$

in $\mathrm{AdS}_5$. Thus a half-BPS operator with $\Delta=p$ maps to a scalar mode satisfying

$$m^2L^2=p(p-4).$$

For example,

$$p=2: \qquad m^2L^2=-4,$$

which saturates the Breitenlohner—Freedman bound in $\mathrm{AdS}_5$.

12. Stress-tensor multiplet

The stress-tensor multiplet is the most important multiplet in any SCFT. In $\mathcal N=4$ SYM, its bottom component is a half-BPS scalar in the $20'$ representation,

$$\mathcal O_2^{IJ} = \operatorname{Tr}(\phi^I\phi^J) - \frac16\delta^{IJ}\operatorname{Tr}(\phi^K\phi^K),$$

with

$$\Delta=2, \qquad SU(4)_R\text{ representation }[0,2,0].$$

Acting with supercharges produces the $R$-symmetry currents, the supersymmetry currents, and the stress tensor:

$$\mathcal O_2 \quad \xrightarrow{\ Q\ } \quad J_\mu^R,\ S_{\mu\alpha} \quad \xrightarrow{\ Q\ } \quad T_{\mu\nu} \quad \text{and other descendants}.$$

In AdS/CFT this single CFT multiplet packages several bulk massless fields:

$$T_{\mu\nu} \leftrightarrow g_{\mu\nu}, \qquad J_\mu^{SO(6)} \leftrightarrow A_\mu^{SO(6)}, \qquad S_{\mu\alpha} \leftrightarrow \psi_{\mu\alpha}.$$

13. Superconformal indices and protected counting

A superconformal index is a trace over the Hilbert space on $S^{d-1}$ that counts states annihilated by a chosen supercharge. Schematically,

$$\mathcal I = \operatorname{Tr}_{S^{d-1}} (-1)^F e^{-\beta\{Q,Q^\dagger\}} \prod_i x_i^{q_i}.$$

Only states satisfying

$$\{Q,Q^\dagger\}=0$$

contribute. Long multiplets cancel between bosons and fermions, while protected short multiplets can survive. Because of this cancellation, the index is invariant under continuous deformations that preserve the chosen supercharge.

The index is powerful, but it is not the full spectrum. It counts protected states with signs and fugacities; it usually cannot tell you the full degeneracy of long multiplets.

14. Supersymmetric Ward identities

Supersymmetry gives Ward identities just like ordinary global symmetry. If $Q$ is conserved and the vacuum is supersymmetric,

$$Q|0\rangle=0,$$

then for local operators $\mathcal O_i$,

$$\left\langle \sum_i (-1)^{F_1+\cdots+F_{i-1}} \mathcal O_1\cdots [Q,\mathcal O_i\}\cdots \mathcal O_n \right\rangle =0.$$

Here $[\, ,\,\}$ is the graded commutator. These Ward identities relate correlators of different components in the same supermultiplet. In a superconformal theory, they strongly constrain two- and three-point functions of protected multiplets.

In AdS/CFT, these identities become relations among bulk couplings inside the same supergravity multiplet.

15. Quick dictionary for AdS/CFT

CFT supersymmetric object	Bulk interpretation
$R$-symmetry group	internal isometry of compact space
$R$-current $J_\mu^a$	bulk gauge field $A_\mu^a$
stress-tensor multiplet	graviton supermultiplet
half-BPS chiral primary	Kaluza—Klein supergravity mode
long single-trace multiplet	massive string state, generally
superconformal index	protected bulk state count
conformal manifold	moduli of AdS vacua or exactly marginal boundary conditions
Wilson line preserving SUSY	fundamental string or brane ending on boundary
defect preserving SUSY	brane embedding with worldvolume $\mathrm{AdS}$ factor

The protected half-BPS tower is the cleanest bridge between $\mathcal N=4$ SYM and type IIB supergravity:

$$\operatorname{Tr}(Y\cdot\phi)^p \quad \longleftrightarrow \quad \text{degree }p\text{ harmonic on }S^5.$$

16. Common traps

A few mistakes show up constantly.

First, do not confuse a supersymmetry multiplet with a conformal family. A supermultiplet contains several conformal primaries related by $Q$. Each conformal primary then has its own $P_\mu$ descendants.

Second, do not assume that a classically marginal operator is exactly marginal. Exact marginality is a statement about vanishing beta functions and operator mixing. Supersymmetry helps, but it does not make every classical marginal deformation exactly marginal.

Third, do not assume that all protected quantities are coupling independent in the same way. A BPS dimension may be fixed, while certain OPE coefficients can still depend on exactly marginal couplings unless additional nonrenormalization theorems apply.

Fourth, do not identify single-trace with protected. Many single-trace operators in $\mathcal N=4$ SYM are long and unprotected.

Fifth, do not identify BPS with free. BPS operators exist at strong coupling and are often the best-controlled observables precisely because they are representation-theoretic, not perturbative.

17. Minimal formula sheet

Supercharges:

$$\{Q^I_\alpha,\bar Q_{J\dot\alpha}\} = 2\delta^I{}_J\sigma^\mu_{\alpha\dot\alpha}P_\mu.$$

Superconformal adjoint relation in radial quantization:

$$Q^\dagger \sim S.$$

Shortening mechanism:

$$\|Q|\mathcal O\rangle\|^2 = \langle\mathcal O|\{S,Q\}|\mathcal O\rangle \geq 0.$$

$\mathcal N=1$ chiral primary:

$$\bar Q_{\dot\alpha}\mathcal O=0, \qquad \Delta=\frac32 R.$$

$\mathcal N=4$ half-BPS tower:

$$\mathcal O_p=\operatorname{Tr}(Y\cdot\phi)^p, \qquad Y^2=0, \qquad \Delta=p, \qquad [0,p,0].$$

AdS scalar relation in $\mathrm{AdS}_{d+1}$:

$$m^2L^2=\Delta(\Delta-d).$$

For $\mathrm{AdS}_5$:

$$m^2L^2=\Delta(\Delta-4).$$

Stress-tensor normalization in four-dimensional $\mathcal N=4$ $SU(N)$ SYM:

$$a=c=\frac{N^2-1}{4}.$$

At large $N$,

$$a=c\sim \frac{N^2}{4}.$$

18. Exercises

Exercise 1

Use the algebraic fact $\{Q,\bar Q\}\sim P$ to explain why the scaling dimension of $Q$ is $\frac12$.

Solution

In a conformal theory, $P_\mu$ has scaling dimension $1$ because it generates translations and raises the dimension of an operator by one:

$$[D,P_\mu]=iP_\mu.$$

The supersymmetry algebra says schematically

$$\{Q,\bar Q\}\sim P.$$

Therefore the dimension of the product $Q\bar Q$ must be $1$. Since $Q$ and $\bar Q$ have the same dimension,

$$[Q]+[\bar Q]=1,$$

so

$$[Q]=[\bar Q]=\frac12.$$

Equivalently, in the superconformal algebra,

$$[D,Q]=\frac{i}{2}Q.$$

Exercise 2

Let $\mathcal O$ be a superconformal primary. Explain why positivity of $\|Q|\mathcal O\rangle\|^2$ leads to BPS bounds.

Solution

In radial quantization, $Q^\dagger$ is proportional to a special superconformal charge $S$. Therefore

$$\|Q|\mathcal O\rangle\|^2 = \langle\mathcal O|Q^\dagger Q|\mathcal O\rangle \sim \langle\mathcal O|S Q|\mathcal O\rangle.$$

For a superconformal primary,

$$S|\mathcal O\rangle=0.$$

Hence

$$S Q|\mathcal O\rangle = \{S,Q\}|\mathcal O\rangle.$$

The anticommutator $\{S,Q\}$ is a linear combination of $D$, Lorentz generators, and $R$-symmetry generators. Acting on a primary with definite quantum numbers, it becomes a number depending on

$$\Delta,\quad \text{spin},\quad R\text{-charges}.$$

Positivity of the norm demands that this number is nonnegative. This gives a unitarity or BPS bound. If the bound is saturated, then

$$\|Q|\mathcal O\rangle\|^2=0,$$

so the descendant $Q|\mathcal O\rangle$ is null. The multiplet shortens.

Exercise 3

For a half-BPS operator $\mathcal O_p$ in $\mathcal N=4$ SYM with $\Delta=p$, compute the AdS$_5$ scalar mass $m^2L^2$ for $p=2,3,4$.

Solution

The scalar mass-dimension relation in $\mathrm{AdS}_5$ is

$$m^2L^2=\Delta(\Delta-4).$$

For the half-BPS tower,

$$\Delta=p.$$

Therefore

$$m^2L^2=p(p-4).$$

For $p=2$,

$$m^2L^2=2(2-4)=-4.$$

For $p=3$,

$$m^2L^2=3(3-4)=-3.$$

For $p=4$,

$$m^2L^2=4(4-4)=0.$$

The $p=2$ mode saturates the Breitenlohner—Freedman bound in $\mathrm{AdS}_5$.

Exercise 4

Explain why the Konishi operator is not dual to a light supergravity field at strong coupling.

Solution

The Konishi operator is a singlet scalar operator schematically of the form

$$\mathcal K = \operatorname{Tr}(\phi^I\phi^I).$$

It is not in a protected half-BPS multiplet. It belongs to a long multiplet, so its scaling dimension is not fixed by the superconformal algebra. As the ‘t Hooft coupling $\lambda$ changes, its dimension can acquire a large anomalous contribution.

In AdS/CFT, light supergravity fields correspond to single-trace operators whose dimensions remain of order one at strong coupling. Protected half-BPS operators are the canonical examples. Long unprotected operators generally become massive string states when $\lambda$ is large. Thus the Konishi operator is not part of the low-energy supergravity spectrum; it is a stringy state.

Exercise 5

Show that the stress-tensor multiplet primary of $\mathcal N=4$ SYM is traceless in its $SO(6)_R$ vector indices.

Solution

The stress-tensor multiplet primary is

$$\mathcal O_2^{IJ} = \operatorname{Tr}(\phi^I\phi^J) - \frac16\delta^{IJ}\operatorname{Tr}(\phi^K\phi^K).$$

Contracting with $\delta^{IJ}$ gives

$$\delta^{IJ}\mathcal O_2^{IJ} = \operatorname{Tr}(\phi^I\phi^I) - \frac16\delta^{IJ}\delta^{IJ}\operatorname{Tr}(\phi^K\phi^K).$$

Since $I,J=1,\ldots,6$,

$$\delta^{IJ}\delta^{IJ}=6.$$

Therefore

$$\delta^{IJ}\mathcal O_2^{IJ} = \operatorname{Tr}(\phi^I\phi^I) - \operatorname{Tr}(\phi^K\phi^K) = 0.$$

Thus $\mathcal O_2^{IJ}$ is symmetric traceless, transforming in the $20'$ representation of $SO(6)_R\simeq SU(4)_R$.