N=4 Super-Yang-Mills

The main idea

The boundary theory in the canonical AdS/CFT example is four-dimensional $\mathcal N=4$ super-Yang-Mills theory with gauge group $SU(N)$. It is the low-energy interacting theory on $N$ coincident D3-branes after the decoupled center-of-mass $U(1)$ sector is removed.

This theory is special for three closely related reasons.

First, it is a four-dimensional conformal field theory with a Lagrangian. The coupling does not run, so the theory exists as a continuous family of CFTs labeled by the complexified gauge coupling

$$\tau_{\mathrm{YM}} = \frac{\theta}{2\pi} + \frac{4\pi i}{g_{\mathrm{YM}}^2}.$$

Second, it is maximally supersymmetric without gravity. In four-dimensional notation it has $16$ Poincare supercharges and, because it is conformal, $16$ conformal supercharges. Together with the conformal group and R-symmetry, these generate the superconformal algebra

$$\mathfrak{psu}(2,2|4).$$

Third, it is a large-$N$ matrix CFT. Gauge-invariant single-trace operators behave like single-particle bulk fields, multi-trace operators behave like multiparticle states, and large-$N$ factorization is the field-theory origin of semiclassical bulk physics.

The elementary fields of $\mathcal N=4$ $SU(N)$ super-Yang-Mills form one adjoint vector multiplet. The six scalars transform as the vector of $SO(6)_R\simeq SU(4)_R$, matching the rotations of the six transverse directions to the D3-branes and the isometries of $S^5$ in the dual geometry.

This page is not trying to teach supersymmetric gauge theory from scratch. The goal is to isolate exactly what an AdS/CFT user needs to know about $\mathcal N=4$ SYM: its fields, action, symmetries, parameters, operator spectrum, and large-$N$ dictionary.

The theory from the D3-brane viewpoint

A single D3-brane fills four directions and sits at a point in six transverse directions. The massless open-string modes on the brane are:

Mode	Four-dimensional field	Meaning
open-string vector polarization along the brane	$A_\mu$	gauge field on the D3-brane
transverse oscillations	$\Phi^I$, $I=1,\ldots,6$	position of the brane in $\mathbb R^6$
fermionic partners	$\lambda^A_\alpha$, $A=1,\ldots,4$	superpartners required by type IIB supersymmetry

For $N$ coincident D3-branes, Chan-Paton labels turn these fields into $N\times N$ matrices. The full low-energy gauge group is $U(N)$, but the overall $U(1)$ describes the free center-of-mass motion of the brane stack. The interacting theory used in AdS/CFT is usually the $SU(N)$ part:

$$U(N)\simeq \frac{SU(N)\times U(1)}{\mathbb Z_N}, \qquad \text{interacting sector: }SU(N).$$

The six scalar matrices are especially important. If they acquire mutually commuting expectation values, their eigenvalues describe the positions of the individual D3-branes in the transverse $\mathbb R^6$. The conformal point dual to $\mathrm{AdS}_5\times S^5$ is the point where all scalar expectation values vanish and the branes are coincident.

Field content

In four-dimensional $\mathcal N=4$ notation, the interacting theory contains the following fields, all in the adjoint representation of $SU(N)$:

Field	Spin	Multiplicity	$SU(4)_R$ representation	On-shell degrees of freedom per color
$A_\mu$	$1$	$1$	${\bf 1}$	$2$
$\lambda^A_\alpha$	$1/2$	$4$ Weyl fermions	${\bf 4}$	$8$ real fermionic
$\Phi^I$	$0$	$6$ real scalars	${\bf 6}$	$6$

The bosonic on-shell count is

$$2+6=8,$$

and the fermionic on-shell count is

$$4\times 2=8.$$

The equality is the first quick check that the fields can form a supersymmetric multiplet. The deeper statement is that this is the unique four-dimensional gauge multiplet with maximal supersymmetry and no spin larger than one.

The R-symmetry has two useful descriptions:

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

The six real scalars transform as the vector ${\bf 6}$ of $SO(6)_R$, or equivalently the antisymmetric representation of $SU(4)_R$. The four Weyl fermions transform as the fundamental ${\bf 4}$ of $SU(4)_R$. In the D3-brane construction, this $SO(6)_R$ is simply the rotation group of the six directions transverse to the branes. In the gravity dual, it becomes the isometry group of $S^5$:

$$SO(6)_R \quad\longleftrightarrow\quad \mathrm{Isom}(S^5).$$

The action, without drowning in indices

A compact way to write the theory is to start from ten-dimensional $\mathcal N=1$ super-Yang-Mills and dimensionally reduce to four dimensions. Let $M,N=0,\ldots,9$ and let $A_M$ be a ten-dimensional gauge field. Reducing on six flat directions means that the fields do not depend on $x^4,\ldots,x^9$, so

$$A_M \longrightarrow (A_\mu,\Phi^I), \qquad \mu=0,\ldots,3, \qquad I=1,\ldots,6.$$

The ten-dimensional field strength contains

$$F_{\mu\nu}, \qquad F_{\mu I}=D_\mu \Phi^I, \qquad F_{IJ}=-i[\Phi^I,\Phi^J],$$

where the factor of $i$ depends on whether the Lie-algebra generators are taken Hermitian or anti-Hermitian. This immediately produces the schematic bosonic action

$$S_{\mathrm{bos}} = \frac{1}{g_{\mathrm{YM}}^2} \int d^4x\, \operatorname{Tr} \left[ -\frac14 F_{\mu\nu}F^{\mu\nu} -\frac12 D_\mu\Phi^I D^\mu\Phi^I +\frac14[\Phi^I,\Phi^J]^2 \right] + \frac{\theta}{8\pi^2} \int \operatorname{Tr}F\wedge F.$$

The full theory also contains fermion kinetic terms and Yukawa couplings of the schematic form

$$\operatorname{Tr} \left( i\bar\lambda \gamma^\mu D_\mu\lambda + \bar\lambda \Gamma_I[\Phi^I,\lambda] \right).$$

The relative coefficients are not optional. Supersymmetry fixes the gauge interactions, scalar quartic interactions, and Yukawa interactions in terms of one coupling $g_{\mathrm{YM}}$.

A useful way to remember the interactions is:

$$\text{interactions vanish on the Coulomb branch} \quad \Longleftrightarrow \quad [\Phi^I,\Phi^J]=0.$$

When the scalar matrices commute, they can be diagonalized simultaneously. Their eigenvalues are the separated D3-brane positions.

Convention warning: where the factors of $2\pi$ hide

The D3-brane literature uses several normalizations for $g_{\mathrm{YM}}$. In this course we follow the convention already used in the D3-brane page:

$$g_{\mathrm{YM}}^2=4\pi g_s, \qquad \lambda=g_{\mathrm{YM}}^2N=4\pi g_sN.$$

Then the type IIB axio-dilaton and the Yang-Mills coupling combine naturally as

$$\tau_{\mathrm{IIB}} = C_0+i e^{-\phi} \quad\longleftrightarrow\quad \tau_{\mathrm{YM}} = \frac{\theta}{2\pi} + \frac{4\pi i}{g_{\mathrm{YM}}^2},$$

with the identification

$$C_0=\frac{\theta}{2\pi}, \qquad e^\phi=g_s=\frac{g_{\mathrm{YM}}^2}{4\pi}.$$

Other authors absorb factors of $2\pi$ differently. Do not compare formulas for $g_{\mathrm{YM}}^2$, $L^4/\alpha'^2$, or $\tau$ across references unless you first check the normalization of the Yang-Mills action and the trace.

The robust physical statements are:

$$\frac{L^4}{\alpha'^2}\sim \lambda, \qquad g_s\sim \frac{\lambda}{N}, \qquad \frac{L^3}{G_5}\sim N^2.$$

The next two pages will fix the detailed version of these relations in the conventions of the course.

Conformal invariance

Classically, the Yang-Mills coupling is dimensionless in four dimensions. Quantum mechanically, most gauge theories develop a scale through renormalization. $\mathcal N=4$ SYM is exceptional: its beta function vanishes.

At one loop, the coefficient of the Yang-Mills beta function for adjoint matter is proportional to

$$b_0 = \frac{11}{3}C_A - \frac{2}{3}n_{\mathrm{Weyl}}C_A - \frac{1}{6}n_{\mathrm{real}}C_A.$$

For $\mathcal N=4$ SYM,

$$n_{\mathrm{Weyl}}=4, \qquad n_{\mathrm{real}}=6,$$

so

$$b_0 = \left( \frac{11}{3} - \frac{8}{3} - 1 \right)C_A =0.$$

Supersymmetry strengthens this one-loop cancellation into exact conformal invariance. The theory therefore has no intrinsic mass scale on $\mathbb R^{1,3}$ at the conformal point. The only scales in observables come from sources, temperature, chemical potential, compactification radii, operator insertions, or chosen states.

This is why the canonical dual is pure $\mathrm{AdS}_5\times S^5$ rather than a geometry with a running radial profile. A running coupling would correspond to a nontrivial radial dependence of bulk scalar fields.

Superconformal symmetry

The bosonic conformal group in four-dimensional Minkowski space is

$$SO(2,4).$$

The R-symmetry is

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

Including supersymmetry, the full superconformal group is

$$PSU(2,2|4),$$

whose bosonic subgroup is

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

On the gravity side,

$$SO(2,4) \quad\longleftrightarrow\quad \mathrm{Isom}(\mathrm{AdS}_5),$$

and

$$SO(6) \quad\longleftrightarrow\quad \mathrm{Isom}(S^5).$$

Thus the spacetime symmetries of the bulk geometry already know about the conformal and R-symmetries of the boundary theory.

This symmetry matching is one of the cleanest first tests of the canonical duality:

$$\mathrm{AdS}_5\times S^5 \quad\text{has isometry}\quad SO(2,4)\times SO(6),$$

while $\mathcal N=4$ SYM has bosonic global symmetry

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

The match is not decorative. It determines the representation theory of protected operators and organizes the Kaluza-Klein spectrum on $S^5$.

The exactly marginal coupling and S-duality

The complex coupling

$$\tau = \frac{\theta}{2\pi} + \frac{4\pi i}{g_{\mathrm{YM}}^2}$$

is exactly marginal. Moving $\tau$ changes the CFT but does not break conformal invariance. In the canonical AdS/CFT example, this continuous conformal manifold maps to the constant type IIB axio-dilaton.

The theory is also expected to have an $SL(2,\mathbb Z)$ duality action

$$\tau \longrightarrow \frac{a\tau+b}{c\tau+d}, \qquad \begin{pmatrix} a & b\\ c & d \end{pmatrix} \in SL(2,\mathbb Z),$$

together with the appropriate action on line operators and, more carefully, on the global form of the gauge group. The generator

$$S:\tau\mapsto -\frac{1}{\tau}$$

exchanges electric and magnetic descriptions. In type IIB string theory, the same $SL(2,\mathbb Z)$ is the nonperturbative duality acting on the axio-dilaton and on fundamental/D-string charges.

For most classical supergravity calculations, S-duality is not the first thing one uses. But conceptually it is a powerful warning: $\mathcal N=4$ SYM is not merely a weakly coupled gauge theory with many symmetries. It is a nonperturbative quantum theory with multiple equivalent descriptions.

Large $N$ and the ‘t Hooft coupling

The two parameters most often used in holography are

$$N, \qquad \lambda=g_{\mathrm{YM}}^2N.$$

The ‘t Hooft large-$N$ limit is

$$N\to\infty, \qquad \lambda\ \text{fixed}.$$

In perturbation theory, each planar diagram contributes a function of $\lambda$, and nonplanar diagrams are suppressed by powers of $1/N^2$. Holographically,

$$\frac{1}{N^2} \quad\text{counts bulk loops},$$

while

$$\frac{1}{\sqrt{\lambda}} \quad\text{counts stringy curvature corrections}$$

in the canonical $\mathrm{AdS}_5\times S^5$ background.

The classical Einstein-gravity limit requires both

$$N\gg 1, \qquad \lambda\gg 1.$$

This is one of the most important lessons of the canonical example. The field theory is easiest at weak coupling $\lambda\ll1$, while the bulk gravity description is easiest at strong coupling $\lambda\gg1$. The duality is powerful precisely because it relates difficult field-theory questions to easier gravitational ones.

Gauge-invariant local operators

The elementary fields $A_\mu$, $\lambda$, and $\Phi^I$ are not gauge-invariant local observables. Physical local operators are gauge-invariant combinations, especially traces of adjoint fields.

A basic class is

$$\mathcal O_k^{I_1\cdots I_k} = \operatorname{Tr} \left( \Phi^{(I_1}\cdots\Phi^{I_k)} \right) -\text{traces},$$

where the parentheses indicate symmetrization in $SO(6)_R$ indices and subtracting traces makes the operator transform in a symmetric traceless representation of $SO(6)_R$.

These are chiral primary operators. Their dimensions are protected:

$$\Delta=k.$$

The lowest nontrivial example is

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

which transforms in the ${\bf 20'}$ of $SU(4)_R$ and lies in the same supermultiplet as the stress tensor.

The contrast with the Konishi operator is instructive. The schematic operator

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

is an $SU(4)_R$ singlet and is not protected. Its dimension depends on $\lambda$:

$$\Delta_{\mathcal K} = 2+O(\lambda) \quad (\lambda\ll1),$$

while at strong coupling it becomes a stringy excitation whose dimension grows parametrically. This is an early glimpse of the distinction between supergravity modes and massive string modes.

Single-trace, multi-trace, and bulk particles

For matrix fields, single-trace operators are the natural large-$N$ building blocks:

$$\operatorname{Tr}(\Phi^k), \qquad \operatorname{Tr}(F_{\mu\nu}\Phi^k), \qquad \operatorname{Tr}(\lambda\lambda\Phi^k), \qquad \operatorname{Tr}(F_{\mu\nu}F^{\mu\nu}\Phi^k), \quad \ldots$$

Very schematically,

$$\text{single-trace primary} \quad\longleftrightarrow\quad \text{single-particle bulk field}.$$

Products of traces correspond to multiparticle states:

$$\mathcal O_1\mathcal O_2 \quad\longleftrightarrow\quad \text{two-particle bulk state}.$$

At infinite $N$, multi-trace dimensions are approximately additive:

$$\Delta_{\mathcal O_1\mathcal O_2} = \Delta_{\mathcal O_1} + \Delta_{\mathcal O_2} + O\!\left(\frac{1}{N^2}\right).$$

This is the CFT version of the statement that weakly interacting particles in the bulk have energies that add.

The word “single-trace” is most natural for $SU(N)$ or $U(N)$ matrix theories. In a more abstract CFT, one should say “single-particle” or “single-string” operators. But for $\mathcal N=4$ SYM, the trace language is both literal and useful.

Stress tensor and central charges

Every CFT has a stress tensor $T_{\mu\nu}$ with protected dimension

$$\Delta_T=d=4.$$

In $\mathcal N=4$ SYM, the stress tensor belongs to a protected supermultiplet whose bottom component is the dimension-two chiral primary $\mathcal O_2^{IJ}$ above.

The four-dimensional Weyl anomaly on a curved background takes the schematic form

$$\langle T^\mu{}_\mu\rangle = \frac{c}{16\pi^2}W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma} - \frac{a}{16\pi^2}E_4 +\cdots,$$

where $E_4$ is the Euler density and the dots denote possible background-gauge-field terms and scheme-dependent total derivatives.

For $\mathcal N=4$ SYM with gauge algebra $\mathfrak{su}(N)$,

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

in the common convention where a free real scalar has

$$a_{\mathrm{scalar}}=\frac{1}{360}, \qquad c_{\mathrm{scalar}}=\frac{1}{120}.$$

The large-$N$ scaling

$$a\sim c\sim N^2$$

is the field-theory origin of

$$\frac{L^3}{G_5}\sim N^2.$$

In holography, $C_T$, $a$, $c$, and $L^3/G_5$ are different normalizations of the same central idea: the number of stress-tensor degrees of freedom is of order $N^2$.

The Coulomb branch

The scalar potential is proportional to

$$V(\Phi) \propto \operatorname{Tr} [\Phi^I,\Phi^J]^2,$$

with the sign chosen so that the energy is nonnegative. Its minima satisfy

$$[\Phi^I,\Phi^J]=0 \qquad \text{for all } I,J.$$

For Hermitian scalar matrices, commuting matrices can be diagonalized simultaneously:

$$\Phi^I = \operatorname{diag} \left( x_1^I,\ldots,x_N^I \right),$$

up to gauge transformations and subject to the tracelessness condition for $SU(N)$. The eigenvalue vector

$$\vec x_a=(x_a^1,\ldots,x_a^6)$$

is the position of the $a$-th D3-brane in the transverse $\mathbb R^6$.

For $U(N)$, the classical moduli space is

$$\mathcal M_{U(N)} = \frac{(\mathbb R^6)^N}{S_N},$$

where the symmetric group permutes identical branes. For $SU(N)$, the center-of-mass coordinate is removed:

$$\sum_{a=1}^N \vec x_a=0.$$

The conformal vacuum dual to $\mathrm{AdS}_5\times S^5$ sits at the origin of this moduli space:

$$\vec x_1=\cdots=\vec x_N=0.$$

Moving onto the Coulomb branch breaks the gauge group and changes the bulk geometry. It should not be confused with changing the state of the same exact $\mathrm{AdS}_5\times S^5$ background.

Line operators and the global form of the gauge group

For many local-operator calculations, it is enough to say “gauge group $SU(N)$.” For line operators and S-duality, the global form matters.

The Lie algebra $\mathfrak{su}(N)$ can correspond to different global gauge groups, such as

$$SU(N), \qquad SU(N)/\mathbb Z_N,$$

with different allowed Wilson and ‘t Hooft line operators. In the string dual, Wilson lines are represented by fundamental strings ending on the boundary, while magnetic line operators are related to D-strings. The $SL(2,\mathbb Z)$ duality rotates these line operators.

This subtlety is not needed for the first derivation of the correspondence, but it becomes important in serious discussions of S-duality, one-form symmetries, confinement diagnostics, and the precise classification of boundary conditions.

Why $\mathcal N=4$ SYM is not QCD

It is tempting to think of $\mathcal N=4$ SYM as “QCD but supersymmetric.” That slogan is too crude.

Feature	$\mathcal N=4$ SYM	QCD-like theory
Matter	adjoint scalars and fermions	fundamental quarks, gluons
Supersymmetry	maximal	none in real QCD
Coupling	exactly marginal	runs with scale
Vacuum at origin	conformal	confining/chiral symmetry breaking in QCD
Degrees of freedom at large $N$	$O(N^2)$ adjoint	gluons $O(N^2)$, quarks often $O(N)$ in the ‘t Hooft limit
Best classical gravity regime	large $N$, large $\lambda$	no known exact classical gravity dual for real QCD

The point of $\mathcal N=4$ SYM is not realism. It is control. It is an exactly defined interacting CFT that admits a string-theoretic dual and, in a strong-coupling large-$N$ regime, a classical gravitational description.

Many later holographic models deform or generalize this example to study confinement, flavor, finite density, hydrodynamics, and condensed-matter-like phenomena. But the canonical example is the calibration point. It teaches which results are universal consequences of gravity, which depend on supersymmetry or conformality, and which are merely model-building assumptions.

The dictionary preview

The next pages will build the bulk side and the precise parameter map. For now, the basic entries are:

$\mathcal N=4$ SYM	Type IIB on $\mathrm{AdS}_5\times S^5$
$SU(N)$ rank	$N$ units of five-form flux through $S^5$
$SO(2,4)$ conformal symmetry	$\mathrm{AdS}_5$ isometry
$SO(6)_R\simeq SU(4)_R$	$S^5$ isometry
$\tau_{\mathrm{YM}}$	type IIB axio-dilaton $\tau_{\mathrm{IIB}}$
$\lambda=g_{\mathrm{YM}}^2N$	string tension in AdS units, $L^2/\alpha'\sim \sqrt\lambda$
$1/N^2$	bulk quantum-loop expansion
single-trace operators	single-particle/string states
chiral primaries	Kaluza-Klein supergravity modes on $S^5$
stress tensor $T_{\mu\nu}$	graviton in $\mathrm{AdS}_5$
R-current $J_\mu^{[IJ]}$	gauge fields from $S^5$ isometries

This table is deliberately schematic. The exact coefficients and normalization conventions are the subject of the next two pages.

Common mistakes

Mistake 1: Treating $\mathcal N=4$ SYM as weakly coupled in the gravity regime.
The classical gravity limit requires $\lambda\gg1$, where ordinary perturbation theory in $g_{\mathrm{YM}}$ is not useful, even though $g_{\mathrm{YM}}^2=\lambda/N$ may be small at large $N$.

Mistake 2: Forgetting the decoupled $U(1)$.
The D3-brane worldvolume theory is naturally $U(N)$, but the interacting AdS/CFT dual usually refers to the $SU(N)$ sector. The overall $U(1)$ is a free center-of-mass multiplet.

Mistake 3: Confusing elementary fields with local CFT observables.
The scalar $\Phi^I$ is not a gauge-invariant local operator. Operators like $\operatorname{Tr}(\Phi^I\Phi^J)$ are.

Mistake 4: Thinking all operator dimensions are protected.
Chiral primaries and conserved currents have protected dimensions. Generic single-trace operators, such as the Konishi operator, acquire anomalous dimensions and become heavy at strong coupling.

Mistake 5: Ignoring the global form of the gauge group.
For local single-trace correlators, the Lie algebra often suffices. For line operators, S-duality, one-form symmetries, and precise nonperturbative statements, the global form matters.

Mistake 6: Equating conformality with triviality.
The theory is conformal for all $\tau$, but it is still interacting. A CFT can have nontrivial anomalous dimensions, OPE coefficients, thermal physics, and transport.

Exercises

Exercise 1: On-shell degrees of freedom

Show that the $\mathcal N=4$ vector multiplet has equal bosonic and fermionic on-shell degrees of freedom per generator of the gauge algebra.

Solution

A massless four-dimensional gauge field has two physical polarizations. The six real scalars contribute six bosonic degrees of freedom. Hence

$$n_{\mathrm{bos}}=2+6=8.$$

A four-dimensional Weyl fermion has two real on-shell degrees of freedom. There are four Weyl fermions, so

$$n_{\mathrm{ferm}}=4\times 2=8.$$

Thus the on-shell bosonic and fermionic degrees of freedom match:

$$n_{\mathrm{bos}}=n_{\mathrm{ferm}}=8.$$

All fields are adjoint-valued, so this count is multiplied by $\dim SU(N)=N^2-1$ for the interacting $SU(N)$ theory.

Exercise 2: One-loop beta-function cancellation

Using

$$b_0 = \frac{11}{3}C_A - \frac{2}{3}n_{\mathrm{Weyl}}C_A - \frac{1}{6}n_{\mathrm{real}}C_A,$$

show that the one-loop beta function vanishes for $\mathcal N=4$ SYM.

Solution

For $\mathcal N=4$ SYM, all matter fields are in the adjoint representation, so they carry the same group factor $C_A$ as the gauge boson. There are

$$n_{\mathrm{Weyl}}=4, \qquad n_{\mathrm{real}}=6.$$

Therefore

$$b_0 = \left( \frac{11}{3} - \frac{2}{3}\cdot 4 - \frac{1}{6}\cdot 6 \right)C_A.$$

This is

$$b_0 = \left( \frac{11}{3} - \frac{8}{3} - 1 \right)C_A = 0.$$

The one-loop cancellation is a quick perturbative signal of conformality. In the full theory, extended supersymmetry implies exact vanishing of the beta function.

Exercise 3: The Coulomb-branch moduli space

Assume the scalar fields $\Phi^I$ are Hermitian matrices and the potential is minimized when

$$[\Phi^I,\Phi^J]=0 \qquad \text{for all }I,J.$$

Explain why the classical moduli space for the $U(N)$ theory is

$$\mathcal M_{U(N)} = \frac{(\mathbb R^6)^N}{S_N}.$$

Solution

If the six Hermitian matrices $\Phi^I$ commute pairwise, they can be diagonalized simultaneously by a unitary gauge transformation. We can write

$$\Phi^I = \operatorname{diag} (x_1^I,\ldots,x_N^I).$$

For each eigenvalue label $a=1,\ldots,N$, the six numbers

$$\vec x_a=(x_a^1,\ldots,x_a^6)$$

define a point in $\mathbb R^6$. Thus a generic commuting configuration describes $N$ points in $\mathbb R^6$.

The remaining Weyl group of $U(N)$ permutes the eigenvalues. Since the D3-branes are identical, configurations related by permutations are gauge-equivalent. Therefore

$$\mathcal M_{U(N)} = \frac{(\mathbb R^6)^N}{S_N}.$$

For $SU(N)$, one removes the center-of-mass coordinate by imposing

$$\sum_{a=1}^N \vec x_a=0.$$

Exercise 4: Central charges from free fields

Use the free-field anomaly coefficients

$$a_s=\frac{1}{360}, \qquad a_f=\frac{11}{720}, \qquad a_v=\frac{31}{180}$$

for a real scalar, Weyl fermion, and vector field, respectively. Show that one $\mathcal N=4$ vector multiplet has

$$a=\frac14.$$

Then infer the value for gauge algebra $\mathfrak{su}(N)$.

Solution

One $\mathcal N=4$ vector multiplet contains one vector, four Weyl fermions, and six real scalars. Therefore

$$a_{\mathcal N=4} = a_v+4a_f+6a_s.$$

Substituting the given values,

$$a_{\mathcal N=4} = \frac{31}{180} + 4\cdot\frac{11}{720} + 6\cdot\frac{1}{360}.$$

With denominator $720$,

$$a_{\mathcal N=4} = \frac{124}{720} + \frac{44}{720} + \frac{12}{720} = \frac{180}{720} = \frac14.$$

For gauge algebra $\mathfrak{su}(N)$, this is multiplied by

$$\dim SU(N)=N^2-1.$$

Thus

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

Supersymmetry also gives

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

Exercise 5: Which operators are light at strong coupling?

Classify the following schematic single-trace operators as protected or generally unprotected:

$$\operatorname{Tr}\!\left(\Phi^{(I}\Phi^{J)}-\text{traces}\right), \qquad \operatorname{Tr}(\Phi^I\Phi^I), \qquad T_{\mu\nu}, \qquad \operatorname{Tr}\Phi^{(I_1}\cdots\Phi^{I_k)}-\text{traces}.$$

Explain what this means for the bulk dual at $\lambda\gg1$.

Solution

The symmetric traceless scalar operators

$$\operatorname{Tr}\!\left(\Phi^{(I}\Phi^{J)}-\text{traces}\right)$$

and more generally

$$\operatorname{Tr}\Phi^{(I_1}\cdots\Phi^{I_k)}-\text{traces}$$

are chiral primaries. Their dimensions are protected, with

$$\Delta=k.$$

The stress tensor $T_{\mu\nu}$ is also protected because it is conserved, and every conserved stress tensor in a four-dimensional CFT has

$$\Delta_T=4.$$

The Konishi operator

$$\operatorname{Tr}(\Phi^I\Phi^I)$$

is generally unprotected. Its dimension depends on the coupling and becomes large at strong ‘t Hooft coupling.

In the bulk dual, protected operators correspond to light supergravity modes or their Kaluza-Klein excitations on $S^5$. Unprotected generic single-trace operators correspond to stringy states whose masses in AdS units grow when $\lambda\gg1$.

Further reading

• J. Maldacena, The Large $N$ Limit of Superconformal Field Theories and Supergravity.
• O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, Large $N$ Field Theories, String Theory and Gravity.
• S. Kovacs, $\mathcal N=4$ Supersymmetric Yang-Mills Theory and the AdS/SCFT Correspondence.
• E. D’Hoker and D. Freedman, Supersymmetric Gauge Theories and the AdS/CFT Correspondence.
• N. Beisert et al., Review of AdS/CFT Integrability: An Overview, for the planar integrability side of $\mathcal N=4$ SYM.