Learning Guide: ABJM Theory and AdS_4/CFT_3

This page is a reference for learning ABJM theory after you have learned the basics of $\mathcal N=4$ SYM and AdS/CFT. It aims to answer:

• What is ABJM theory, exactly?
• Why does it describe M2-branes?
• Why is it conformal in 3d, and why do we need Chern–Simons terms?
• What are monopole operators, and why are they essential?
• How does the dual gravity description work (M-theory vs type IIA)?
• What are the standard limits, parameters, and “sanity checks” of the duality?

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0. Big picture: ABJM is to M2-branes what $\mathcal N=4$ SYM is to D3-branes

If $\mathcal N=4$ SYM is your reference point, keep this analogy in mind:

• D3-branes (type IIB): the low-energy worldvolume theory is 4d $\mathcal N=4$ SYM, and at large $N$ it is dual to type IIB string theory on $AdS_5\times S^5$.

• M2-branes (M-theory): the low-energy worldvolume theory should be a 3d superconformal field theory (SCFT). ABJM theory provides a concrete Lagrangian description for $N$ M2-branes on the orbifold $\mathbb C^4/\mathbb Z_k$, and at large $N$ it is dual to M-theory on $AdS_4\times S^7/\mathbb Z_k$ (or type IIA on $AdS_4\times CP^3$ in an appropriate limit).

Two qualitative differences from the D3/$\mathcal N=4$ SYM story are worth emphasizing early:

1. Dimension matters: in 3d, an ordinary Yang–Mills term is not conformal (its coupling has dimensions), so you need a different gauge-field “engine” to get an interacting fixed point. That engine is Chern–Simons theory.

2. Degrees of freedom scale differently: at strong coupling, observables often scale like $N^{3/2}$ (characteristic of M2-branes), rather than $N^2$ (characteristic of D3-branes).

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1. Prerequisites, conventions, and a quick glossary

What you should already know (minimal)

If you know the basic ideas of $\mathcal N=4$ SYM and AdS/CFT, you are in good shape. The key extra ingredients for ABJM are:

• Chern–Simons gauge theory in 3d (very different from 4d Yang–Mills),
• 3d superconformal symmetry (what “$\mathcal N=6$” means in 3d),
• monopole (disorder) operators in 3d CFTs.

Conventions and notation

• Spacetime dimension: ABJM lives in 2+1 dimensions (one time, two space).
• Gauge group and CS level: we will write the ABJM gauge group as $$U(N)_k\times U(N)_{-k},$$ meaning the first factor has Chern–Simons level $+k$ and the second has level $-k$.
• Matter fields: there are two common notations.
  • $Y^A$ notation: four complex scalars $Y^A$ ($A=1,2,3,4$) and fermions $\psi_A$ transforming under an $SU(4)$ $R$-symmetry.
  • $(A_i, B_j)$ notation: two doublets $A_1,A_2$ and $B_1,B_2$ (convenient in $\mathcal N=2$ superspace). Roughly, $Y^A$ packages these into an $SU(4)$-covariant object.
• “$\mathcal N$” in 3d: in 3d, $\mathcal N$ counts real supercharges in units of a 2-component Majorana spinor.
  • 3d $\mathcal N=6$ means 12 real supercharges.
  • 3d $\mathcal N=8$ means 16 real supercharges (maximal).

Glossary (quick)

• CS term: topological gauge-field term $\sim \int \mathrm{Tr}(A\wedge dA + \frac{2}{3}A^3)$.
• Level $k$: quantized integer coefficient of the CS term; in ABJM it also labels the orbifold $\mathbb C^4/\mathbb Z_k$.
• SCFT: superconformal field theory (a CFT with supersymmetry).
• Monopole operator: a local operator in 3d defined by inserting a quantized magnetic flux through an $S^2$ surrounding the insertion point.
• Topological $U(1)$ symmetry: a global symmetry whose current is $*\mathrm{Tr}(F)$ in 3d.
• Hopf fibration: $S^7$ viewed as an $S^1$ fiber over $CP^3$; the $\mathbb Z_k$ orbifold acts along the $S^1$ fiber.

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2. Why Chern–Simons in 3d: the “conformal gauge field” in 2+1 dimensions

2.1 Yang–Mills is not conformal in 3d

In 4d, the Yang–Mills coupling $g_{\text{YM}}$ is dimensionless, which is why conformal invariance is plausible and (for $\mathcal N=4$) exact.

In 3d, the Yang–Mills action

$$S_{\text{YM}}\sim \frac{1}{g_{\text{YM}}^2}\int d^3x\,\mathrm{Tr}(F_{\mu\nu}F^{\mu\nu})$$

has $[g_{\text{YM}}^2] = \text{mass}$, so it introduces a scale. A pure 3d Yang–Mills theory is not conformal; it flows strongly in the IR and typically confines.

2.2 Chern–Simons is classically scale-invariant

The Chern–Simons action in 3d is

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

Key points:

• $k$ must be an integer (level quantization).
• The CS gauge field has no local propagating degrees of freedom by itself (it is “topological”).
• When coupled to matter, it mediates interactions and produces nontrivial dynamics while remaining compatible with conformal symmetry.

2.3 Parity and the need for opposite levels

In 2+1 dimensions, a parity transformation reverses orientation and flips the sign of the CS term:

$$S_{\text{CS}} \to -S_{\text{CS}}.$$

ABJM achieves parity invariance by using two gauge fields with opposite levels:

$$S_{\text{CS}}[A] - S_{\text{CS}}[\hat A].$$

Under parity, you swap $A\leftrightarrow \hat A$, restoring invariance.

This “two-node quiver with opposite levels” is a hallmark of ABJM.

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3. The ABJM field theory: gauge group, matter content, and what the Lagrangian looks like

3.1 Definition (the data that defines the theory)

ABJM theory is defined by:

• Gauge group: $$U(N)_k\times U(N)_{-k}.$$
• Matter: four complex scalars and four complex fermions in bifundamental representations.

In the $Y^A$ notation:

• Scalars $Y^A$ transform in $(\mathbf N,\overline{\mathbf N})$,
• Conjugates $Y_A^\dagger$ transform in $(\overline{\mathbf N},\mathbf N)$,
• Fermions $\psi_A$ are superpartners.

The global (manifest) $R$-symmetry is

$$SU(4)_R \cong SO(6)_R,$$

under which $Y^A$ transforms as a $\mathbf 4$.

There is also a crucial topological global symmetry in 3d (see Section 5), which is tied to monopole operators and M-theory circle momentum.

3.2 A useful $\mathcal N=2$ superspace viewpoint

Many people first meet ABJM in 3d $\mathcal N=2$ language (because it looks like a familiar quiver gauge theory):

• Two vector multiplets for $U(N)$ and $U(N)$.
• Four chiral multiplets:
  • $A_1, A_2$ in $(\mathbf N,\overline{\mathbf N})$,
  • $B_1, B_2$ in $(\overline{\mathbf N},\mathbf N)$.
• A quartic superpotential (schematically) $$W \sim \frac{2\pi}{k}\,\epsilon^{ij}\epsilon^{\dot i\dot j}\, \mathrm{Tr}(A_i B_{\dot i} A_j B_{\dot j}).$$

This is a good “entry point” because it makes it clear that:

• the CS level $k$ plays the role of a coupling,
• the scalar potential ends up being sextic (roughly $\sim (2\pi/k)^2\,\mathrm{Tr}(|Y|^6)$) after integrating out auxiliary fields.

3.3 What the component Lagrangian contains (conceptual structure)

At the component level, the ABJM Lagrangian has:

1. Chern–Simons terms for $A_\mu$ and $\hat A_\mu$ at levels $+k$ and $-k$.
2. Kinetic terms for matter: $$\int d^3x\,\mathrm{Tr}\big( D_\mu Y_A^\dagger D^\mu Y^A + i\,\bar\psi^A \gamma^\mu D_\mu \psi_A\big).$$
3. Yukawa couplings fixed by supersymmetry.
4. A sextic scalar potential whose minima give the moduli space in Section 4.

You usually do not need the full explicit component Lagrangian to start learning ABJM, but you should know what pieces are there and why the potential is sextic: in 3d CS-matter theories, the gauge field is not propagating in the same way as in 4d YM, and the supersymmetric completion forces interaction terms that are effectively higher-order in the scalars.

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4. Moduli space and why ABJM describes M2-branes on $\mathbb C^4/\mathbb Z_k$

A decisive consistency check is that the space of vacua matches the expected transverse geometry of M2-branes.

4.1 Warm-up: the abelian case $N=1$

Take $U(1)_k\times U(1)_{-k}$. The matter fields are just four complex scalars $Y^A$.

Define the linear combinations of gauge fields

$$A_\pm = \frac{1}{2}(A \pm \hat A).$$

• The matter fields couple only to $A_-$ (because they have opposite charges under the two $U(1)$’s).
• The CS terms reduce schematically to a mixed form $\sim \frac{k}{2\pi}\int A_-\wedge dA_+$.

A standard way to think about this:

• $A_+$ acts as a Lagrange multiplier enforcing that $A_-$ is (locally) flat.
• Large gauge transformations and CS level quantization imply that a flat $A_-$ still has discrete holonomy.
• That discrete holonomy acts on $Y^A$ as a phase identification $$Y^A \sim e^{2\pi i/k}\,Y^A.$$

Therefore the moduli space for $N=1$ is

$$\mathcal M_{N=1} \cong \mathbb C^4/\mathbb Z_k.$$

Interpretation:

• For $k=1$, this is just $\mathbb C^4$ (flat transverse space of a single M2-brane in 11d Minkowski).
• For $k>1$, it is an orbifold singularity, matching the geometry an M2-brane probes in the ABJM setup.

4.2 Non-abelian case: $N$ indistinguishable branes

For general $N$, the scalar fields are matrices. Vacua are configurations where the scalar potential vanishes, which essentially forces the matrices to be simultaneously diagonalizable up to gauge transformations (heuristically: the vacuum wants commuting scalars).

The resulting moduli space is the symmetric product:

$$\mathcal M_{N} \cong \mathrm{Sym}^N(\mathbb C^4/\mathbb Z_k) = (\mathbb C^4/\mathbb Z_k)^N/S_N.$$

Interpretation:

• The $N$ eigenvalues correspond to positions of $N$ M2-branes in $\mathbb C^4/\mathbb Z_k$.
• The quotient by $S_N$ reflects that the branes are indistinguishable.

This is the direct analog of the $\mathcal N=4$ SYM statement that the Coulomb branch moduli space is $(\mathbb R^6)^N/S_N$ for $N$ D3-branes.

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5. Monopole operators: the “new ingredient” in 3d CFTs

If you only know $\mathcal N=4$ SYM, this is the single biggest conceptual jump.

5.1 What is a monopole operator?

In a 3d CFT, a local operator $\mathcal M(x)$ can be defined not only as a polynomial of fields at $x$ (a “normal” operator), but also as a disorder operator defined by a boundary condition around $x$.

A monopole operator is defined by demanding that on a small $S^2$ surrounding the insertion point,

$$\int_{S^2} \frac{F}{2\pi} = m,$$

where $m$ is an integer (or, for non-abelian groups, a set of integers specifying flux in the Cartan).

Using the state–operator correspondence in CFT$_3$:

• inserting $\mathcal M(0)$ corresponds to a state on $S^2$ with magnetic flux $m$ through the sphere.

5.2 Why CS theories make monopoles subtle (and important)

In a pure CS theory, Gauss’ law ties electric charge to magnetic flux. Roughly,

$$J^\mu \sim \frac{k}{2\pi}\,\varepsilon^{\mu\nu\rho}F_{\nu\rho}.$$

Consequences:

• A “bare” monopole operator typically carries gauge charge (it transforms in a representation of the gauge group), because turning on flux induces an electric charge in CS theory.
• To build a gauge-invariant local operator, you often must dress a monopole operator with matter fields.

This is not an optional technicality: it is structurally how the operator spectrum of ABJM works.

5.3 The topological $U(1)$ and why it matches M-theory circle momentum

In 3d, for each $U(1)$ gauge field (or each $U(N)$ with its trace), there is a conserved topological current

$$j_T^\mu = \frac{1}{2\pi}\varepsilon^{\mu\nu\rho}\,\mathrm{Tr}(F_{\nu\rho}), \qquad \partial_\mu j_T^\mu = 0.$$

• The corresponding charge counts magnetic flux through spatial $S^2$.
• In ABJM’s dual geometry, this charge is naturally identified with momentum along the M-theory circle (the Hopf fiber of $S^7/\mathbb Z_k$).

A useful “physics picture”:

• Operators built only from $Y^A$ and $\psi_A$ typically correspond to supergravity modes with zero momentum along the M-theory circle.
• To get states with momentum, you need monopole charge in the field theory.

5.4 Baryon-like / wrapped-brane operators

ABJM has gauge-invariant operators that look “baryonic,” but unlike in 4d they often require monopole dressing to be gauge-invariant.

On the gravity side these map to wrapped branes:

• D4-branes wrapping cycles in $CP^3$ (type IIA regime),
• M5-branes wrapping cycles in $S^7/\mathbb Z_k$ (M-theory regime).

Understanding these operators is an excellent way to learn how “non-perturbative objects” in the bulk appear as operators in the boundary theory.

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6. Special cases $k=1,2$: supersymmetry enhancement to $\mathcal N=8$

ABJM is constructed with manifest 3d $\mathcal N=6$ supersymmetry (12 supercharges). But for $k=1$ and $k=2$, the theory is believed (and extensively checked) to have an enhanced $\mathcal N=8$ symmetry in the IR.

What changes?

• Geometrically, $\mathbb C^4/\mathbb Z_k$ preserves fewer supersymmetries for generic $k$, but for $k=1,2$ the orbifold action is special enough that the full $SO(8)$ symmetry can re-emerge.
• In field theory, the enhancement is not manifest in the UV Lagrangian. The missing currents and supercharges appear through monopole operators that complete the multiplets into full $SO(8)$ representations.

A practical takeaway:

• If you want the “cleanest” M2-brane theory (flat transverse space), focus on $k=1$.
• If you want the simplest orbifold that still has $\mathcal N=8$, focus on $k=2$.
• For generic $k$, ABJM is the canonical $\mathcal N=6$ example and is often technically simpler in controlled limits (e.g. large $k$ for weak coupling).

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7. Couplings and limits: what plays the role of $g_{\text{YM}}$ and $\lambda$?

7.1 Discrete coupling: the CS level $k$

Unlike $\mathcal N=4$ SYM, ABJM does not have a continuously tunable gauge coupling. The CS level $k$ is an integer.

Still, at large $N$ there is a natural continuous parameter:

7.2 ’t Hooft coupling

Define

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

• The planar (large-$N$) perturbative expansion is organized in powers of $\lambda$.
• Small $\lambda$ corresponds (roughly) to large $k$ at fixed $N$.
• Large $\lambda$ corresponds to strong coupling.

This is the analog of $\lambda=g_{\text{YM}}^2N$ in $\mathcal N=4$ SYM.

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8. The holographic dual: M-theory on $AdS_4\times S^7/\mathbb Z_k$ and type IIA on $AdS_4\times CP^3$

8.1 The geometry and the role of the orbifold

A key geometric fact:

• $S^7$ can be written as an $S^1$ Hopf fibration over $CP^3$.
• The $\mathbb Z_k$ quotient acts by shifting the Hopf fiber angle, effectively shrinking the $S^1$ by a factor of $k$.

Therefore:

• For small $k$, the circle is not small and the right description is M-theory.
• For large $k$, the circle becomes small and you can reduce to type IIA string theory on $CP^3$.

8.2 Parameter map (scaling relations you should remember)

Let $N$ be the rank and $k$ the CS level.

• In the M-theory regime, the AdS radius in 11d Planck units scales as

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

  So classical 11d supergravity becomes accurate when $Nk\gg 1$.

• In the type IIA regime, the AdS radius in string units scales like

  $$\frac{R^2}{\alpha'} \sim \sqrt{\lambda}.$$

  So stringy curvature corrections are small when $\lambda\gg 1$.

• The type IIA string coupling scales as

  $$g_s \sim \frac{\lambda^{5/4}}{N} \quad (\text{up to numerical constants}).$$

  So string loops are suppressed at large $N$ (planar limit), as expected.

These relations are the ABJM analogs of the familiar $AdS_5\times S^5$ scalings $R^4/\alpha'^2\sim\lambda$ and $g_s\sim\lambda/N$.

8.3 Two standard holographic regimes

It’s useful to keep the regimes separate:

1. M-theory regime: $N\to\infty$ with $k$ fixed (or not too large).
   Dual: M-theory on $AdS_4\times S^7/\mathbb Z_k$.
   This is where the $N^{3/2}$ scaling of degrees of freedom is most transparent.

2. Type IIA / planar regime: $N,k\to\infty$ with $\lambda=N/k$ fixed.
   Dual: type IIA string theory on $AdS_4\times CP^3$.
   Classical string theory is valid when $\lambda\gg 1$, and string loops are suppressed when $N\gg 1$.

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9. Benchmark checks and “things to compute first”

When you learn ABJM, it’s helpful to have a few “anchor computations” that appear again and again.

9.1 Free energy on $S^3$ and the $N^{3/2}$ law

A celebrated result (computed via supersymmetric localization) is that at large $N$ in the M-theory regime, the sphere free energy scales as

$$F_{S^3} = -\log Z_{S^3} \sim N^{3/2}\,\sqrt{k} \quad (\text{up to a known numerical factor}).$$

Interpretation:

• This matches the gravitational expectation for the number of degrees of freedom of $N$ M2-branes.
• It is one of the cleanest quantitative confirmations that ABJM is the correct M2-brane SCFT.

Contrast:

• In $\mathcal N=4$ SYM, many observables scale as $N^2$ at large $N$ (D3-brane degrees of freedom).

9.2 Wilson loops

ABJM has multiple types of supersymmetric Wilson loops:

• $1/6$-BPS Wilson loops (more “basic”),
• $1/2$-BPS Wilson loops (require coupling to both gauge fields and matter in a specific way).

Via localization, certain Wilson loop expectation values can be computed exactly (in terms of a matrix model), and at strong coupling they match minimal surfaces of strings in $AdS_4\times CP^3$.

9.3 Operator spectrum: chiral primaries and KK modes

There are protected operators built from $Y^A$ and $Y_A^\dagger$ that form short multiplets of the superconformal algebra. In the gravity dual:

• these correspond to Kaluza–Klein harmonics on $S^7/\mathbb Z_k$ (M-theory) or on $CP^3$ (type IIA).

To see the full KK tower (including momentum along the M-theory circle), you must include monopole operators on the field theory side (Section 5).

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10. How ABJM connects back to $\mathcal N=4$ SYM (practical comparisons)

A quick comparison that helps intuition:

• Spacetime dimension: SYM is 4d, ABJM is 3d.
• Gauge dynamics: SYM uses Yang–Mills kinetic term; ABJM uses Chern–Simons terms (no propagating gauge bosons without matter).
• Couplings: SYM has continuous $g_{\text{YM}}$ and $\theta$; ABJM has discrete $k$, and planar physics depends on $\lambda=N/k$.
• R-symmetry: SYM has $SU(4)_R\cong SO(6)$ exactly; ABJM has manifest $SU(4)_R$, and for $k=1,2$ it enhances to $SO(8)_R$.
• Non-perturbative operators: SYM has instantons and line operators; ABJM has local monopole operators that are central to the spectrum.
• Gravity dual: SYM dual is $AdS_5\times S^5$; ABJM dual is $AdS_4\times S^7/\mathbb Z_k$ or $AdS_4\times CP^3$.

If you remember only one moral:

ABJM is a “minimal” holographic SCFT in 3d whose structure is tightly constrained by supersymmetry, topology (CS level), and the M2-brane geometry.

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11. A structured learning path (syllabus-style)

Below is a practical path that many students find effective. The idea is to build from “what is new in 3d” to “ABJM specifics” to “holography checks.”

Stage A: 3d Chern–Simons and 3d SUSY basics (1–2 weeks)

1. Chern–Simons theory basics

   • Level quantization, gauge invariance, parity.
   • How CS gauge fields differ from YM gauge fields.

2. Matter coupled to CS

   • How CS-matter theories can have interacting fixed points.
   • The meaning of “topological symmetry” in 3d: $j_T\sim *F$.

Recommended reading

• Any good set of 3d CS notes (field theory course notes).
• A short review on CS-matter SCFTs (often included in ABJM lecture notes).

Stage B: ABJM definition and moduli space (1–2 weeks)

3. Read the ABJM construction

   • Understand the gauge group $U(N)_k\times U(N)_{-k}$ and matter fields.
   • Understand why the superpotential is quartic and why the scalar potential is sextic.

4. Work through the abelian moduli space derivation

   • Derive $\mathbb C^4/\mathbb Z_k$ carefully.
   • Then understand the symmetric product for general $N$.

Recommended reading

• Original ABJM paper: O. Aharony, O. Bergman, D. Jafferis, J. Maldacena (2008), arXiv:0806.1218.
• A pedagogical review: e.g. notes titled “M2-branes and $AdS_4/CFT_3$” by leading authors in the field.

Stage C: Monopole operators and supersymmetry enhancement (2–3 weeks)

5. Learn monopole operators in 3d CFT

   • Definition via flux on $S^2$ (state–operator map).
   • How CS terms induce gauge charge for monopoles.
   • Dressing monopoles to form gauge-invariant operators.

6. Understand why $k=1,2$ enhances to $\mathcal N=8$

   • The role of monopole operators in completing $SO(8)$ multiplets.

Recommended reading

• Papers/notes on monopole operators in CS-matter theories.
• Notes focusing specifically on supersymmetry enhancement in ABJM at $k=1,2$.

Stage D: Holography: M-theory vs type IIA limits (2–3 weeks)

7. Geometry

   • $S^7$ as a Hopf fibration over $CP^3$.
   • How $\mathbb Z_k$ acts and why large $k$ gives type IIA.

8. Parameter map

   • Memorize the scaling relations in Section 8.2.
   • Learn what is controlled in each regime (curvature vs loops).

Recommended reading

• AdS$_4$/CFT$_3$ review chapters in string theory lecture notes.
• Sections of ABJM-focused review papers discussing $AdS_4\times CP^3$.

Stage E: “First exact results” via localization (optional but very valuable)

9. Localization in ABJM
   • Sphere partition function $Z_{S^3}$ and free energy scaling.
   • Supersymmetric Wilson loops and their matrix model.

Recommended reading

• Papers by Kapustin, Willett, Yaakov (localization in 3d).
• Reviews by Marino and collaborators on the ABJM matrix model and exact results.

Stage F: Optional advanced topics

• ABJ theory ($U(N)_k\times U(M)_{-k}$, $N\neq M$) and discrete torsion.
• Integrability in planar ABJM (spin chains, spectral problem, quantum spectral curve).
• Higher-point correlators and bootstrap in 3d SCFTs.
• Higher-spin dualities related to vector models vs CS-matter limits.

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12. A final practical checklist

If you can explain these points cleanly, you understand ABJM at a good level:

1. Why 3d YM is not conformal, and why CS is used instead.
2. The field content and gauge group of ABJM, and why opposite levels matter.
3. Why the moduli space is $\mathrm{Sym}^N(\mathbb C^4/\mathbb Z_k)$.
4. What a monopole operator is, why it is often not gauge-invariant by itself, and how to dress it.
5. Why $k=1,2$ is special and leads to $\mathcal N=8$ enhancement.
6. The basic AdS$_4$/CFT$_3$ dual map: when the dual is M-theory on $AdS_4\times S^7/\mathbb Z_k$ vs type IIA on $AdS_4\times CP^3$.
7. At least one quantitative check you trust (e.g. $F_{S^3}\sim N^{3/2}\sqrt{k}$).

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References

• ABJM (2008): Aharony–Bergman–Jafferis–Maldacena, arXiv:0806.1218.
• ABJ (2008): Aharony–Bergman–Jafferis, arXiv:0807.4924 (generalization to $U(N)\times U(M)$).
• BLG models: original Bagger–Lambert and Gustavsson papers (2007–2008) on $\mathcal N=8$ 3-algebra theories.