M2-Branes, ABJM, and AdS4/CFT3

The main idea

The D3-brane example taught us the cleanest form of AdS/CFT:

$$\mathcal N=4\ \mathrm{SYM}_4 \quad\longleftrightarrow\quad \text{type IIB string theory on } \mathrm{AdS}_5\times S^5.$$

M2-branes give the next great example, and it is different in almost every useful way. The boundary theory is three-dimensional rather than four-dimensional, the gravity dual is naturally eleven-dimensional M-theory rather than perturbative string theory, and the number of degrees of freedom scales as $N^{3/2}$ rather than $N^2$.

The central duality is the ABJM correspondence:

$$\boxed{ U(N)_k\times U(N)_{-k}\ \mathcal N=6\ \text{Chern-Simons-matter theory} \quad\longleftrightarrow\quad \text{M-theory on }\mathrm{AdS}_4\times S^7/\mathbb Z_k. }$$

Here $k$ is an integer Chern-Simons level. The same theory has a second large-$N$ limit, the planar ‘t Hooft limit,

$$N\to\infty, \qquad k\to\infty, \qquad \lambda=\frac{N}{k}\ \text{fixed},$$

in which the dual description becomes type IIA string theory on

$$\boxed{ \mathrm{AdS}_4\times \mathbb{CP}^3. }$$

This page has three goals. First, it explains why M2-branes lead naturally to a three-dimensional Chern-Simons-matter CFT rather than an ordinary Yang-Mills theory. Second, it derives the geometric dictionary between $N$, $k$, the M-theory radius, the type IIA coupling, and the curvature scale. Third, it explains the famous $N^{3/2}$ scaling, one of the sharpest lessons that AdS/CFT teaches beyond the matrix-large-$N$ intuition of D-branes.

The ABJM correspondence has two useful gravity languages. At fixed $k$ and large $N$, the natural bulk theory is eleven-dimensional M-theory on $\mathrm{AdS}_4\times S^7/\mathbb Z_k$. In the planar limit with $\lambda=N/k$ fixed, the Hopf fiber of $S^7/\mathbb Z_k$ becomes the M-theory circle, and the dual becomes type IIA string theory on $\mathrm{AdS}_4\times \mathbb{CP}^3$.

The slogan is:

$$\boxed{ \text{M2-branes are not just D3-branes in one fewer dimension.} }$$

The difference is structural. D3-branes give a four-dimensional Yang-Mills theory with a marginal coupling. M2-branes give a three-dimensional SCFT whose natural weakly described Lagrangian uses Chern-Simons gauge fields, bifundamental matter, and discrete level $k$.

Why M2-branes are different from D3-branes

A single M2-brane in flat eleven-dimensional spacetime has a $2+1$-dimensional worldvolume and eight transverse scalar fields. For one brane this is simple: the low-energy theory is a free $\mathcal N=8$ theory of eight scalars and eight fermions. For many coincident M2-branes, however, the interacting theory is much subtler than the D-brane gauge theories discussed earlier.

For D$p$-branes, open strings end on the branes. The massless open-string sector gives a $p+1$ dimensional supersymmetric Yang-Mills theory, with coupling

$$[g_{\mathrm{YM}}^2]=\mathrm{mass}^{3-p}.$$

For D3-branes, $g_{\mathrm{YM}}^2$ is dimensionless. That is why the D3-brane system naturally leads to a conformal gauge theory. For D2-branes, by contrast,

$$[g_{\mathrm{YM}}^2]=\mathrm{mass}.$$

Three-dimensional Yang-Mills theory is not conformal in the ultraviolet or infrared by simple dimensional analysis. It can nevertheless flow to a strongly coupled infrared fixed point. Physically, this infrared fixed point is the theory of M2-branes, because D2-branes lift to M2-branes when the M-theory circle decompactifies.

The question is then:

Can one write a useful Lagrangian for the interacting multiple-M2-brane fixed point itself?

The answer is yes, but not as an ordinary Yang-Mills theory. In three dimensions there is another natural gauge interaction: the Chern-Simons term

$$S_{\mathrm{CS}}[A] = \frac{k}{4\pi} \int \mathrm{tr}\left( A\wedge dA+\frac{2i}{3}A\wedge A\wedge A \right).$$

The coefficient $k$ is dimensionless and quantized. A pure Chern-Simons gauge field has no local propagating degrees of freedom, so coupling it to matter can produce a scale-invariant interacting theory without introducing a dimensionful Yang-Mills coupling. This is the basic reason ABJM theory is a Chern-Simons-matter theory.

The supergravity throat of M2-branes

Before introducing the boundary Lagrangian, let us recall the gravity side. The extremal M2-brane solution of eleven-dimensional supergravity has metric

$$ds_{11}^2 = H(r)^{-2/3}\eta_{\alpha\beta}dx^\alpha dx^\beta + H(r)^{1/3}\left(dr^2+r^2d\Omega_7^2\right), \qquad \alpha=0,1,2,$$

with harmonic function

$$H(r)=1+\frac{R^6}{r^6}.$$

The four-form flux $F_4$ is electrically sourced by the M2-branes. Flux quantization fixes

$$\boxed{ R^6=32\pi^2 N\ell_p^6 }$$

for $N$ M2-branes in flat transverse space $\mathbb R^8$, where $\ell_p$ is the eleven-dimensional Planck length.

In the near-horizon region $r\ll R$,

$$H(r)\simeq \frac{R^6}{r^6},$$

and the metric becomes

$$ds_{11}^2 = \frac{r^4}{R^4}\eta_{\alpha\beta}dx^\alpha dx^\beta + \frac{R^2}{r^2}dr^2 + R^2d\Omega_7^2.$$

After a simple radial redefinition and a rescaling of the boundary coordinates, this is

$$\boxed{ \mathrm{AdS}_4\times S^7, \qquad L_{\mathrm{AdS}_4}=\frac{R}{2}, \qquad L_{S^7}=R. }$$

The factor of $2$ between the $S^7$ radius and the AdS$_4$ radius is not a convention-free detail that can always be ignored. It matters in precise Kaluza-Klein spectra, flux quantization, and comparisons of protected operator dimensions.

The ABJM generalization replaces the transverse space by an orbifold

$$\mathbb R^8\simeq \mathbb C^4 \quad\longrightarrow\quad \mathbb C^4/\mathbb Z_k.$$

The near-horizon geometry becomes

$$\boxed{ \mathrm{AdS}_4\times S^7/\mathbb Z_k. }$$

Because the internal volume is divided by $k$, flux quantization gives

$$\boxed{ R^6=32\pi^2 kN\ell_p^6. }$$

Thus $N$ is the flux through $S^7/\mathbb Z_k$, while $k$ controls the orbifold action and, after reduction, the size of the M-theory circle.

The Hopf fibration and the type IIA limit

The seven-sphere can be viewed as a circle fibration over complex projective space:

$$S^1\longrightarrow S^7\longrightarrow \mathbb{CP}^3.$$

The $\mathbb Z_k$ quotient acts along the Hopf fiber, so the circle becomes shorter:

$$S^1/\mathbb Z_k\longrightarrow S^7/\mathbb Z_k\longrightarrow \mathbb{CP}^3.$$

The radius of the M-theory circle is schematically

$$R_{11}\sim \frac{R}{k}.$$

If this circle is large in eleven-dimensional Planck units, the proper description is M-theory. If this circle is small, one should reduce to type IIA string theory on the base $\mathbb{CP}^3$.

Using

$$\frac{R}{\ell_p}\sim (kN)^{1/6},$$

one finds

$$\frac{R_{11}}{\ell_p} \sim \frac{(kN)^{1/6}}{k} = \left(\frac{N}{k^5}\right)^{1/6}.$$

Therefore:

$$\boxed{ \text{M-theory regime: } N\gg k^5, }$$

while

$$\boxed{ \text{type IIA regime: } N\ll k^5. }$$

In the type IIA regime it is natural to define the ABJM ‘t Hooft coupling

$$\boxed{ \lambda=\frac{N}{k}. }$$

The string-frame curvature radius and string coupling scale as

$$\boxed{ \frac{L^2}{\alpha'}\sim \sqrt{\lambda}, \qquad g_s\sim \frac{\lambda^{5/4}}{N}. }$$

Thus classical type IIA supergravity requires

$$\boxed{ 1\ll \lambda\ll N^{4/5}. }$$

The first inequality suppresses stringy $\alpha'$ corrections. The second suppresses string loops and keeps the M-theory circle small enough for the IIA reduction.

This is already a major conceptual difference from the D3-brane case. In AdS$_5$/CFT$_4$, large $N$ and large $\lambda$ lead directly to weakly curved classical type IIB supergravity. In ABJM theory, there are two distinct strong-coupling gravity regimes:

Limit	Boundary variables	Bulk description	Expansion
M-theory limit	$N\to\infty$, $k$ fixed	M-theory on $\mathrm{AdS}_4\times S^7/\mathbb Z_k$	$1/N$ quantum M-theory corrections
Planar string limit	$N,k\to\infty$, $\lambda=N/k$ fixed	type IIA string theory on $\mathrm{AdS}_4\times \mathbb{CP}^3$	genus expansion plus $\alpha'$ corrections
Classical IIA supergravity	$N\gg 1$, $1\ll\lambda\ll N^{4/5}$	two-derivative type IIA supergravity	small $g_s$ and small curvature
Weak planar ABJM	$N,k\to\infty$, $\lambda\ll 1$	perturbative Chern-Simons-matter	weak planar field theory

The ABJM field theory

ABJM theory is a three-dimensional superconformal Chern-Simons-matter theory with gauge group

$$\boxed{ U(N)_k\times U(N)_{-k}. }$$

The two subscripts denote opposite Chern-Simons levels. The opposite signs are essential: a single Chern-Simons term violates parity, but the ABJM theory has a parity symmetry that combines spatial reflection with exchange of the two gauge fields.

The field content can be summarized as follows:

Field	Representation	Role
$A_\mu$	adjoint of first $U(N)$	Chern-Simons gauge field at level $k$
$\widehat A_\mu$	adjoint of second $U(N)$	Chern-Simons gauge field at level $-k$
$Y^A$, $A=1,\ldots,4$	$(\mathbf N,\overline{\mathbf N})$	four complex scalar fields
$\psi_A$	$(\mathbf N,\overline{\mathbf N})$	fermionic partners
conjugates	$(\overline{\mathbf N},\mathbf N)$	complete bifundamental matter multiplets

The matter fields transform under

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

for generic $k$, and the theory has explicit $\mathcal N=6$ superconformal symmetry. For $k=1,2$, the supersymmetry is enhanced to $\mathcal N=8$, and the full R-symmetry becomes

$$SO(8)_R,$$

as expected for M2-branes in flat space or on the mild $\mathbb C^4/\mathbb Z_2$ orbifold.

A schematic form of the action is

$$S_{\mathrm{ABJM}} = S_{\mathrm{CS}}[A;k] + S_{\mathrm{CS}}[\widehat A;-k] + \int d^3x\,\mathrm{tr}\left( -D_\mu Y_A^\dagger D^\mu Y^A +i\bar\psi^A\gamma^\mu D_\mu\psi_A -V_{\mathrm{sextic}}+\cdots \right).$$

The ellipsis includes Yukawa couplings fixed by supersymmetry. The scalar potential is sextic, not quartic, as required in three dimensions: a scalar has engineering dimension $1/2$, so a classically marginal scalar potential has degree six.

There is no continuous Yang-Mills coupling. The parameter $k$ is discrete. The continuous-looking coupling $\lambda=N/k$ appears only in the large-$N$ planar expansion.

Moduli space and the meaning of $\mathbb C^4/\mathbb Z_k$

A decisive check of ABJM theory is its moduli space. The moduli space should describe the positions of $N$ indistinguishable M2-branes moving in the transverse orbifold $\mathbb C^4/\mathbb Z_k$:

$$\boxed{ \mathcal M_N = \mathrm{Sym}^N\left(\mathbb C^4/\mathbb Z_k\right). }$$

Let us first consider $N=1$. The gauge group is $U(1)_k\times U(1)_{-k}$. The matter fields $Y^A$ are charged under the relative $U(1)$ but neutral under the diagonal $U(1)$. The Chern-Simons terms imply that after gauge identifications the scalar coordinates are subject to

$$Y^A\sim e^{2\pi i/k}Y^A, \qquad A=1,\ldots,4.$$

Thus the moduli space is

$$\mathbb C^4/\mathbb Z_k.$$

For general $N$, one can diagonalize the scalar matrices on the moduli space. The eigenvalues describe $N$ branes, and permutation of eigenvalues by the Weyl group gives the symmetric product.

This simple statement hides one of the characteristic subtleties of three-dimensional gauge theory: monopole operators. In three dimensions, local disorder operators can insert magnetic flux through a small two-sphere surrounding the insertion point. In Chern-Simons theory, magnetic flux carries electric charge because the Chern-Simons equation of motion schematically relates charge and flux:

$$\frac{k}{2\pi}F \sim \star J.$$

Therefore monopole operators in ABJM theory are not optional exotic decorations. They are needed for the full operator spectrum, for the correct moduli-space interpretation, and for supersymmetry enhancement at $k=1,2$.

A useful rule of thumb is:

$$\boxed{ \text{In ABJM, some geometrically obvious bulk quantum numbers are carried by monopole operators.} }$$

For example, momentum along the M-theory circle is not visible as an ordinary polynomial in the elementary bifundamental fields. It is encoded by monopole sectors.

Symmetries and the AdS$_4$/CFT$_3$ dictionary

The bosonic symmetry of a three-dimensional CFT is

$$SO(2,3),$$

which matches the isometry group of $\mathrm{AdS}_4$. The internal symmetry of the round $S^7$ is $SO(8)$, matching the $SO(8)_R$ symmetry of the $\mathcal N=8$ M2-brane theory. After orbifolding by $\mathbb Z_k$, the manifest internal symmetry is reduced to

$$SU(4)\times U(1),$$

matching the generic ABJM global symmetry. In the type IIA description, this is the geometric symmetry of $\mathbb{CP}^3$ together with the remnant associated to the M-theory circle.

The basic dictionary is:

CFT$_3$ object	Bulk object
Stress tensor $T_{\mu\nu}$	AdS$_4$ graviton
$SU(4)_R$ currents	gauge fields from isometries of $\mathbb{CP}^3$ or $S^7/\mathbb Z_k$
Chiral primary operators	Kaluza-Klein modes on the internal space
Monopole operators	states carrying M-theory circle momentum or wrapped-brane charges
Wilson loops	fundamental strings in IIA, or M2-branes wrapping the M-circle in M-theory
$S^3$ partition function $F$	Euclidean AdS$_4$ on-shell action
Thermal state on $\mathbb R^2$	planar AdS$_4$ black brane
Supersymmetric black-hole index	BPS AdS$_4$ black-hole microstate counting

In a three-dimensional CFT there is no ordinary Weyl anomaly on a closed manifold, so one does not use four-dimensional central charges $a$ and $c$. Two especially useful measures of degrees of freedom are:

$$C_T, \qquad F_{S^3}=-\log |Z_{S^3}|.$$

Here $C_T$ normalizes the stress-tensor two-point function, and $F_{S^3}$ is the sphere free energy. In holographic CFT$_3$ theories both scale like

$$\frac{L_{\mathrm{AdS}}^2}{G_4}.$$

For ABJM, this gives the famous scaling

$$\boxed{ F_{S^3}\sim C_T\sim k^{1/2}N^{3/2}. }$$

More precisely, at leading order for ABJM on the round three-sphere,

$$\boxed{ F_{S^3} = \frac{\pi\sqrt{2k}}{3}N^{3/2}+O(N^{1/2}) }$$

in the M-theory large-$N$ regime.

This result is not a minor numerical curiosity. It shows that the fundamental degrees of freedom of coincident M2-branes are not counted by adjoint matrices in the same way as D3-branes. The $N^{3/2}$ law is one of the clearest signatures of M-theoretic holography.

Where the $N^{3/2}$ scaling comes from

Let us derive the scaling from the bulk. The four-dimensional Newton constant is obtained by reducing eleven-dimensional supergravity on $S^7/\mathbb Z_k$:

$$\frac{1}{G_4} \sim \frac{\mathrm{Vol}(S^7/\mathbb Z_k)}{G_{11}} \sim \frac{R^7}{k\ell_p^9}.$$

The AdS$_4$ radius is $L\sim R$, up to the fixed factor $1/2$. Therefore the dimensionless gravitational strength is

$$\frac{L^2}{G_4} \sim \frac{R^9}{k\ell_p^9}.$$

Using flux quantization,

$$\frac{R^6}{\ell_p^6}\sim kN,$$

we obtain

$$\frac{R^9}{k\ell_p^9} \sim \frac{(kN)^{3/2}}{k} = k^{1/2}N^{3/2}.$$

Thus

$$\boxed{ \frac{L^2}{G_4}\sim k^{1/2}N^{3/2}. }$$

For $k=1$, this is simply $N^{3/2}$. Compare this with the D3-brane case:

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

The difference is not merely the boundary dimension. It is a difference between the large-$N$ physics of D-branes and M-branes.

Localization and exact tests

ABJM theory is strongly coupled in the M-theory regime, but supersymmetry makes some exact computations possible. Supersymmetric localization reduces the $S^3$ partition function of ABJM theory to a finite-dimensional matrix model. At large $N$, this matrix model reproduces the gravity prediction

$$F_{S^3}=\frac{\pi\sqrt{2k}}{3}N^{3/2}+\cdots.$$

Even more impressively, the matrix model can be reorganized as a Fermi gas. In that formulation, the large-$N$ expansion has an Airy-function structure, and certain quantum corrections become calculable with remarkable precision.

The conceptual lesson is important for the whole course:

$$\boxed{ \text{AdS/CFT can sometimes compute quantum gravity corrections by exact CFT methods.} }$$

In AdS$_5$/CFT$_4$, the canonical theory $\mathcal N=4$ SYM has many exact tools, especially integrability and localization. ABJM theory has its own exact toolkit: supersymmetric localization, matrix models, Fermi-gas methods, superconformal indices, topologically twisted indices, and protected Wilson loops. These tools have made AdS$_4$/CFT$_3$ one of the best laboratories for studying quantum M-theory.

Comparison with the D3-brane duality

It is useful to compare the canonical examples side by side.

Feature	D3-branes	M2-branes / ABJM
Boundary dimension	$d=4$	$d=3$
Canonical CFT	$\mathcal N=4$ SYM	ABJM Chern-Simons-matter theory
Gauge structure	$SU(N)$ or $U(N)$ Yang-Mills	$U(N)_k\times U(N)_{-k}$ Chern-Simons
Coupling	$\lambda=g_{\mathrm{YM}}^2N$	$\lambda=N/k$ in planar limit
Bulk at strong planar coupling	type IIB on $\mathrm{AdS}_5\times S^5$	type IIA on $\mathrm{AdS}_4\times \mathbb{CP}^3$
Bulk at fixed discrete parameter	still type IIB	M-theory on $\mathrm{AdS}_4\times S^7/\mathbb Z_k$
Degree scaling	$N^2$	$k^{1/2}N^{3/2}$
Internal symmetry	$SO(6)_R$	$SU(4)_R\times U(1)$, enhanced to $SO(8)_R$ for $k=1,2$
Exact tools	integrability, localization	localization, Fermi gas, indices, integrability in planar IIA regime

Two morals are worth isolating.

First, AdS/CFT is not synonymous with gauge theory in four dimensions. It is a broader mechanism relating quantum gravity in asymptotically AdS spaces to nongravitational CFTs in one lower dimension.

Second, the classical-gravity limit is not always a string-theory limit. In ABJM at fixed $k$, the correct large-$N$ bulk is eleven-dimensional M-theory. This is why ABJM is often the cleanest example for asking questions about genuinely M-theoretic quantum gravity.

A quick guide to the ABJM regimes

The following diagnostic is useful in practice.

Start from $N$ and $k$. Define

$$\lambda=\frac{N}{k}.$$

Then ask:

1. Is $N\gg 1$? If not, there is no classical bulk limit.
2. Is $k$ fixed or at least $N\gg k^5$? Then the natural bulk is eleven-dimensional M-theory.
3. Is $N\ll k^5$? Then the M-theory circle is small, so reduce to type IIA.
4. In the type IIA regime, is $\lambda\gg 1$? Then the string-frame curvature is small.
5. Is $\lambda\ll N^{4/5}$? Then the string coupling is small.

This gives:

$$\boxed{ \begin{array}{ccl} N\gg k^5 &\Rightarrow& \text{M-theory on }\mathrm{AdS}_4\times S^7/\mathbb Z_k,\\ 1\ll \lambda\ll N^{4/5} &\Rightarrow& \text{classical type IIA on }\mathrm{AdS}_4\times\mathbb{CP}^3,\\ \lambda\ll 1 &\Rightarrow& \text{weakly coupled planar ABJM perturbation theory.} \end{array} }$$

Do not compress this to “large $\lambda$ means gravity.” Large $\lambda$ by itself does not tell you whether the correct weakly curved description is type IIA or eleven-dimensional M-theory.

Common mistakes

Mistake 1: Treating ABJM as ordinary Yang-Mills. The ABJM gauge fields are Chern-Simons gauge fields. There is no Maxwell kinetic term in the conformal theory. Adding a Yang-Mills term is an irrelevant deformation that can be useful as a regulator or as a route from D2-branes, but it is not part of the fixed-point definition.

Mistake 2: Forgetting that $k$ is quantized. The Chern-Simons level is an integer. The parameter $\lambda=N/k$ becomes continuously useful only in a large-$N$ limit.

Mistake 3: Assuming all large-$N$ holographic CFTs have $N^2$ degrees of freedom. Matrix gauge theories often have $N^2$ scaling, but M2-branes have $N^{3/2}$ scaling. M5-branes, discussed next, have $N^3$ scaling.

Mistake 4: Confusing the M-theory and type IIA limits. The fixed-$k$ large-$N$ regime is not planar string theory. It is an eleven-dimensional regime.

Mistake 5: Ignoring monopole operators. Many important ABJM states and symmetries are invisible if one only writes polynomial traces of elementary fields. Monopole sectors are central to the theory.

Mistake 6: Dropping the radius factor $L_{S^7}=2L_{\mathrm{AdS}}$. For qualitative discussions this often does no harm. For spectra and precision matching, it matters.

Exercises

Exercise 1: Why Chern-Simons, not Yang-Mills?

In three dimensions, determine the mass dimension of $g_{\mathrm{YM}}^2$ in a Yang-Mills action

$$S_{\mathrm{YM}} = \frac{1}{4g_{\mathrm{YM}}^2} \int d^3x\, \mathrm{tr}\,F_{\mu\nu}F^{\mu\nu}.$$

Explain why this makes Yang-Mills theory unsuitable as the fundamental conformal interaction of M2-branes.

Solution

The action is dimensionless and $d^3x$ has mass dimension $-3$. In $d$ dimensions, $[g_{\mathrm{YM}}^2]=\mathrm{mass}^{4-d}$. Therefore in $d=3$,

$$[g_{\mathrm{YM}}^2]=\mathrm{mass}.$$

The Yang-Mills coupling is dimensionful, so it introduces a scale. Three-dimensional Yang-Mills theory can flow to a nontrivial infrared fixed point, but the Yang-Mills term itself is not a marginal conformal interaction. A Chern-Simons term has dimensionless quantized level $k$, so Chern-Simons-matter theory is the natural Lagrangian framework for a three-dimensional gauge-theoretic SCFT.

Exercise 2: The near-horizon M2-brane geometry

Starting from

$$ds_{11}^2 = H^{-2/3}\eta_{\alpha\beta}dx^\alpha dx^\beta + H^{1/3}\left(dr^2+r^2d\Omega_7^2\right), \qquad H=\frac{R^6}{r^6},$$

show that the near-horizon metric is $\mathrm{AdS}_4\times S^7$ with $L_{\mathrm{AdS}_4}=R/2$.

Solution

Substituting $H=R^6/r^6$ gives

$$H^{-2/3}=\frac{r^4}{R^4}, \qquad H^{1/3}=\frac{R^2}{r^2}.$$

Thus

$$ds_{11}^2 = \frac{r^4}{R^4}\eta_{\alpha\beta}dx^\alpha dx^\beta +\frac{R^2}{r^2}dr^2 +R^2d\Omega_7^2.$$

Let

$$y=\frac{r^2}{R^2}.$$

Then

$$\frac{R^2}{r^2}dr^2 = \frac{R^2}{4}\frac{dy^2}{y^2},$$

and the first term is $y^2\eta_{\alpha\beta}dx^\alpha dx^\beta$, up to a constant rescaling of the boundary coordinates. Therefore the four-dimensional part is Poincaré AdS$_4$ with radius $R/2$, while the internal sphere has radius $R$.

Exercise 3: The $N^{3/2}$ scaling from flux quantization

Use

$$R^6\sim kN\ell_p^6, \qquad \frac{1}{G_4}\sim \frac{R^7}{k\ell_p^9}, \qquad L\sim R,$$

to show that

$$\frac{L^2}{G_4}\sim k^{1/2}N^{3/2}.$$

Solution

The four-dimensional gravitational strength scales as

$$\frac{L^2}{G_4} \sim \frac{R^2R^7}{k\ell_p^9} = \frac{R^9}{k\ell_p^9}.$$

From flux quantization,

$$\frac{R}{\ell_p}\sim (kN)^{1/6}.$$

Therefore

$$\frac{R^9}{k\ell_p^9} \sim \frac{(kN)^{9/6}}{k} = \frac{(kN)^{3/2}}{k} = k^{1/2}N^{3/2}.$$

Since holographic CFT$_3$ observables such as $F_{S^3}$ and $C_T$ scale as $L^2/G_4$, they scale as $k^{1/2}N^{3/2}$.

Exercise 4: Classify the bulk regime

For each pair $(N,k)$, determine whether the natural strong-coupling bulk description is M-theory or type IIA, assuming $N$ is large.

1. $N=10^8$, $k=1$.
2. $N=10^8$, $k=10^3$.
3. $N=10^8$, $k=10^7$.

Solution

The key comparison is $N$ versus $k^5$.

1. For $k=1$, $k^5=1$, so $N\gg k^5$. This is the M-theory regime.
2. For $k=10^3$, $k^5=10^{15}$, so $N\ll k^5$. The M-theory circle is small, so the natural description is type IIA. Since $\lambda=N/k=10^5$ and $N^{4/5}=10^{6.4}$, the inequalities $1\ll\lambda\ll N^{4/5}$ are satisfied. This is a clean classical type IIA supergravity regime.
3. For $k=10^7$, $k^5=10^{35}$, so the circle is very small, but $\lambda=N/k=10$. This may be in the type IIA string regime with small string coupling, but the curvature is only moderately small. It is not as clean a classical supergravity limit as case 2.

Exercise 5: The $N=1$ moduli space

Explain why the moduli space of the $U(1)_k\times U(1)_{-k}$ ABJM theory is $\mathbb C^4/\mathbb Z_k$ rather than simply $\mathbb C^4$.

Solution

The four complex scalars $Y^A$ are coordinates on $\mathbb C^4$ before gauge identifications. In the $U(1)_k\times U(1)_{-k}$ theory, the diagonal $U(1)$ decouples from the matter, while the relative $U(1)$ acts by a common phase rotation on the $Y^A$. Because of the Chern-Simons level $k$, the residual gauge identification is a discrete phase rotation

$$Y^A\sim e^{2\pi i/k}Y^A.$$

Therefore the physical moduli space is

$$\mathbb C^4/\mathbb Z_k.$$

For $N$ branes one then obtains the symmetric product $\mathrm{Sym}^N(\mathbb C^4/\mathbb Z_k)$.

Further reading

• O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, $\mathcal N=6$ Superconformal Chern-Simons-Matter Theories, M2-Branes and Their Gravity Duals.
• I. R. Klebanov and G. Torri, M2-Branes and AdS/CFT.
• K. Hosomichi, M2-Branes and AdS/CFT: A Review.
• I. R. Klebanov and A. A. Tseytlin, Entropy of Near-Extremal Black p-Branes.
• A. Kapustin, B. Willett, and I. Yaakov, Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter.
• N. Drukker, M. Mariño, and P. Putrov, From Weak to Strong Coupling in ABJM Theory.
• M. Mariño, Lectures on Localization and Matrix Models in Supersymmetric Chern-Simons-Matter Theories.
• D. Martelli and J. Sparks, Moduli Spaces of Chern-Simons Quiver Gauge Theories and AdS$_4$/CFT$_3$.
• J. Bagger and N. Lambert, Gauge Symmetry and Supersymmetry of Multiple M2-Branes.
• A. Gustavsson, Algebraic Structures on Parallel M2-Branes.