M5-Branes and Six-Dimensional CFTs

The main idea

The M5-brane example is the most mysterious member of the original AdS/CFT triad:

$$\begin{array}{ccl} \text{D3-branes} &\longrightarrow& \mathrm{AdS}_5/\mathrm{CFT}_4,\\ \text{M2-branes} &\longrightarrow& \mathrm{AdS}_4/\mathrm{CFT}_3,\\ \text{M5-branes} &\longrightarrow& \mathrm{AdS}_7/\mathrm{CFT}_6. \end{array}$$

The M5-brane duality is

$$\boxed{ \text{M-theory on }\mathrm{AdS}_7\times S^4 \quad\longleftrightarrow\quad \text{the six-dimensional }(2,0)\text{ SCFT of type }A_{N-1}. }$$

Here $N$ is the number of coincident M5-branes, or equivalently the number of four-form flux units through $S^4$. The interacting boundary theory is not a six-dimensional Yang-Mills theory. It has no known ordinary local Lagrangian, contains tensionless string-like excitations at the conformal point, and has degrees of freedom scaling as $N^3$, not $N^2$.

That last fact is the first reason this page belongs in an AdS/CFT course. The D3-brane example might tempt us to equate holography with matrix large-$N$ counting. M5-branes say: not so fast. The gravitational measure of degrees of freedom is not always $N^2$; for M5-branes it is

$$\frac{L_{\mathrm{AdS}}^5}{G_7}\sim N^3.$$

The second reason is even more important. The six-dimensional $(2,0)$ theory is one of the best examples of a quantum field theory whose existence is strongly supported by string/M-theory, supersymmetry, compactification, anomalies, and holography, but whose intrinsic formulation remains highly nontrivial. It is not merely another item in the AdS/CFT zoo; it is a reminder that AdS/CFT often teaches us what quantum field theory itself can be.

The M5-brane correspondence. The near-horizon geometry of $N$ coincident M5-branes is $\mathrm{AdS}_7\times S^4$, with $N$ units of $F_4$ flux through $S^4$. The dual CFT is the interacting part of the six-dimensional $(2,0)$ theory of type $A_{N-1}$. Its compactifications generate many lower-dimensional theories, including five-dimensional maximally supersymmetric Yang-Mills, four-dimensional $\mathcal N=4$ SYM, and class-$\mathcal S$ theories.

The slogan is:

$$\boxed{ \text{The }(2,0)\text{ theory is a parent of gauge theories, not merely a strange cousin.} }$$

Four-dimensional $\mathcal N=4$ SYM, its S-duality, five-dimensional maximally supersymmetric Yang-Mills, and broad families of four-dimensional $\mathcal N=2$ theories are most naturally understood as compactifications of this six-dimensional object.

From M5-branes to $\mathrm{AdS}_7\times S^4$

An M5-brane is a five-dimensional spatial brane in eleven-dimensional M-theory. Its worldvolume has signature $5+1$, and its transverse space has dimension five. The single M5-brane worldvolume theory contains a free $(2,0)$ tensor multiplet:

$$\text{one chiral two-form } B, \qquad H=dB, \qquad H=*_{6}H,$$

plus five scalar fields and fermions fixed by supersymmetry. The five scalars describe transverse fluctuations of the brane and transform as a vector of $SO(5)_R$, the rotation group of the transverse space.

For $N$ coincident M5-branes, there is an interacting six-dimensional superconformal theory. The center-of-mass motion gives a decoupled free tensor multiplet; the interacting sector is usually called the $A_{N-1}$ $(2,0)$ theory. More generally, $(2,0)$ SCFTs are labeled by simply laced Lie algebras of ADE type:

$$\mathfrak g=A_{N-1},D_N,E_6,E_7,E_8.$$

The $A_{N-1}$ case is the one directly obtained from $N$ coincident M5-branes in flat space and is the case dual to the simplest $\mathrm{AdS}_7\times S^4$ background.

The eleven-dimensional supergravity solution sourced by $N$ coincident extremal M5-branes takes the form

$$ds_{11}^2 = H(r)^{-1/3}\eta_{\alpha\beta}dx^\alpha dx^\beta + H(r)^{2/3}\left(dr^2+r^2d\Omega_4^2\right), \qquad \alpha=0,1,\ldots,5,$$

with

$$H(r)=1+\frac{R^3}{r^3}, \qquad R^3=\pi N\ell_p^3.$$

The solution also carries magnetic four-form flux through the transverse $S^4$,

$$\frac{1}{(2\pi\ell_p)^3}\int_{S^4}F_4=N,$$

up to the standard convention-dependent normalization of $F_4$.

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

$$H(r)\simeq \frac{R^3}{r^3},$$

so the metric becomes

$$ds_{11}^2 = \frac{r}{R}\eta_{\alpha\beta}dx^\alpha dx^\beta + \frac{R^2}{r^2}dr^2 + R^2d\Omega_4^2.$$

Introduce a Poincare coordinate $z$ by

$$r=\frac{4R^3}{z^2}.$$

Then

$$\frac{r}{R}dx_{1,5}^2+\frac{R^2}{r^2}dr^2 = \frac{4R^2}{z^2}\left(dz^2+dx_{1,5}^2\right).$$

Thus the near-horizon geometry is

$$\boxed{ \mathrm{AdS}_7\times S^4, \qquad L_{\mathrm{AdS}}=2R, \qquad L_{S^4}=R. }$$

Equivalently, if $L\equiv L_{\mathrm{AdS}}$, then

$$\boxed{ L^3=8\pi N\ell_p^3, \qquad L_{S^4}=\frac{L}{2}. }$$

This is the M5-brane analog of the D3-brane derivation of $\mathrm{AdS}_5\times S^5$. The structure is the same, but the parameter dependence is different. In type IIB on $\mathrm{AdS}_5\times S^5$, the curvature radius is controlled by $\lambda^{1/4}$ in string units. In M-theory on $\mathrm{AdS}_7\times S^4$, there is no string coupling and no string length. There is only the eleven-dimensional Planck length $\ell_p$, and

$$\frac{L}{\ell_p}\sim N^{1/3}.$$

The classical supergravity limit is therefore

$$N\gg 1.$$

There is no independent ‘t Hooft coupling to tune.

Why the boundary theory is six-dimensional

The conformal boundary of $\mathrm{AdS}_7$ is six-dimensional. In Poincare coordinates,

$$ds_{\mathrm{AdS}_7}^2 = \frac{L^2}{z^2}\left(dz^2+\eta_{\alpha\beta}dx^\alpha dx^\beta\right), \qquad \alpha=0,1,\ldots,5,$$

and the boundary lies at $z=0$. The boundary theory therefore lives on $\mathbb R^{1,5}$, or after conformal compactification on

$$\mathbb R\times S^5.$$

The isometry group of $\mathrm{AdS}_7$ is

$$SO(2,6),$$

which is precisely the conformal group in six spacetime dimensions. The isometry group of the internal sphere is

$$SO(5),$$

which becomes the $R$-symmetry group of the $(2,0)$ SCFT. The full superconformal symmetry is

$$\boxed{ OSp(8^*|4), \qquad \text{bosonic subgroup }SO(2,6)\times SO(5)_R. }$$

This symmetry match is one of the cleanest pieces of the dictionary:

Bulk object	Boundary object
$\mathrm{AdS}_7$ isometries $SO(2,6)$	six-dimensional conformal symmetry
$S^4$ isometries $SO(5)$	$SO(5)_R$ symmetry
$N$ units of $F_4$ flux	rank/type $A_{N-1}$ data
11d gravitons on $\mathrm{AdS}_7\times S^4$	stress-tensor multiplet and protected operators
M2-branes ending on the boundary	surface operators / self-dual strings
M5-branes wrapping cycles in $S^4$ or AdS	extended defects and wrapped-brane observables

Notice the difference from D3-branes. In four-dimensional $\mathcal N=4$ SYM, the gauge field is one of the elementary degrees of freedom. In the six-dimensional $(2,0)$ theory, the elementary free tensor multiplet contains a two-form potential $B$, not a one-form gauge field $A$. For the interacting nonabelian theory, even the phrase “elementary field” is dangerous. Many observables are more naturally associated with strings and surfaces than with particles and Wilson lines.

The $(2,0)$ tensor multiplet and the interacting theory

For a single M5-brane, the free $(2,0)$ tensor multiplet contains:

Field	Interpretation	$SO(5)_R$ representation
$B_{\alpha\beta}$ with $H=dB=*H$	chiral two-form gauge potential	singlet
$\phi^I$, $I=1,\ldots,5$	transverse fluctuations	vector $\mathbf 5$
fermions	supersymmetric partners	spinor of $SO(5)_R$

The self-duality condition is crucial:

$$H_{\alpha\beta\gamma} = \frac{1}{3!}\epsilon_{\alpha\beta\gamma\delta\epsilon\zeta} H^{\delta\epsilon\zeta}.$$

It halves the degrees of freedom of the two-form, much as chirality halves the degrees of freedom of a Weyl fermion. It also makes covariant Lagrangian formulations subtle even for the free theory.

For multiple M5-branes, the interacting theory is not obtained by simply replacing $U(1)$ by $U(N)$ in a two-form gauge theory. There is no known ordinary nonabelian two-form gauge theory that captures the full interacting $(2,0)$ theory in a manifestly local six-dimensional Lagrangian. The theory is instead characterized by a network of mutually consistent facts:

• it exists as the low-energy theory of coincident M5-branes;
• it has $OSp(8^*|4)$ superconformal symmetry;
• it has ADE classification;
• compactification on $S^1$ gives five-dimensional maximally supersymmetric Yang-Mills;
• compactification on $T^2$ gives four-dimensional $\mathcal N=4$ SYM with S-duality;
• compactification on Riemann surfaces gives class-$\mathcal S$ theories;
• holography predicts large-$N$ observables from M-theory on $\mathrm{AdS}_7\times S^4$.

That is why the $(2,0)$ theory is often treated as a definition by its consequences. This sounds unsatisfying at first, but it is powerful. The theory is overconstrained from many sides.

The $N^3$ degrees of freedom

The most famous quantitative statement about $N$ M5-branes is

$$\boxed{ \text{number of degrees of freedom} \sim N^3. }$$

Holographically, the reason is short. Reduce eleven-dimensional supergravity on $S^4$. If $L$ is the AdS$_7$ radius, then the $S^4$ radius is $L/2$, so

$$\frac{1}{G_7} = \frac{\mathrm{Vol}(S^4_{L/2})}{G_{11}} = \frac{1}{G_{11}}\frac{8\pi^2}{3}\left(\frac{L}{2}\right)^4.$$

Using

$$G_{11}=16\pi^7\ell_p^9, \qquad L^3=8\pi N\ell_p^3,$$

one obtains

$$\boxed{ \frac{L^5}{G_7}=\frac{16}{3\pi^2}N^3. }$$

This is the six-dimensional analog of the familiar AdS$_5$/CFT$_4$ result

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

But the physical meaning is more surprising. In ordinary adjoint matrix theories, $N^2$ suggests fields with two color indices. The $(2,0)$ theory has no conventional adjoint Lagrangian, and the $N^3$ scaling is not explained by a simple matrix count. It is instead a robust gravitational fact.

A clean thermodynamic manifestation comes from the planar AdS$_7$ black brane:

$$ds_7^2 = \frac{L^2}{z^2}\left[-f(z)dt^2+d\vec x_5^{\,2}+\frac{dz^2}{f(z)}\right], \qquad f(z)=1-\left(\frac{z}{z_h}\right)^6.$$

The temperature is

$$T=\frac{6}{4\pi z_h}=\frac{3}{2\pi z_h},$$

and the entropy density is

$$s = \frac{1}{4G_7}\left(\frac{L}{z_h}\right)^5 = \frac{L^5}{4G_7}\left(\frac{2\pi T}{3}\right)^5.$$

Using the formula above,

$$\boxed{ s= \frac{128\pi^3}{729}N^3T^5. }$$

The $T^5$ follows from six-dimensional conformal invariance. The $N^3$ is the nontrivial information.

The same scaling appears in the holographic Weyl anomaly of the six-dimensional theory. In even-dimensional CFTs, the trace of the stress tensor on a curved background has local curvature-anomaly terms. For the M5-brane theory, holographic renormalization gives anomaly coefficients that grow as $N^3$ at large $N$.

There are several useful ways to remember this:

$$\boxed{ \begin{array}{ccl} \text{M2-branes} &:& F_{S^3}\sim N^{3/2},\\ \text{D3-branes} &:& a,c,s/T^3\sim N^2,\\ \text{M5-branes} &:& a_{6d},C_T,s/T^5\sim N^3. \end{array} }$$

The powers are not arbitrary. They come from flux quantization and dimensional reduction of the gravitational action.

The supergravity regime

For the M5-brane duality, the parameter controlling the classical bulk approximation is

$$\frac{\ell_p}{L}\sim N^{-1/3}.$$

Thus large $N$ suppresses eleven-dimensional quantum-gravity corrections and higher-derivative corrections. Schematically,

$$S_{\mathrm{11d}} = \frac{1}{\ell_p^9}\int d^{11}x\sqrt g\,R + \frac{1}{\ell_p^3}\int C_3\wedge F_4\wedge F_4 + \text{higher-derivative terms}.$$

The familiar eleven-dimensional $R^4$-type higher-derivative terms are suppressed by powers of

$$\left(\frac{\ell_p}{L}\right)^6 \sim \frac{1}{N^2}.$$

Bulk loops are also suppressed at large $N$. In the boundary theory this corresponds to a large-$N$ expansion, but it is not the usual planar expansion of a four-dimensional matrix gauge theory.

A useful comparison is:

Duality	Curvature control	Loop control	Classical bulk limit
D3 / $\mathcal N=4$ SYM	$L^4/\alpha'^2\sim\lambda$	$g_s\sim\lambda/N$	$N\gg 1$, $\lambda\gg 1$
M2 / ABJM at fixed $k$	$L/\ell_p\sim N^{1/6}$	same 11d expansion	$N\gg 1$
M5 / $(2,0)$	$L/\ell_p\sim N^{1/3}$	same 11d expansion	$N\gg 1$

For M5-branes there is no parameter analogous to the four-dimensional ‘t Hooft coupling. The strongly coupled nature of the boundary theory is built in.

Operators and Kaluza-Klein modes

Local operators of the six-dimensional SCFT are organized by $OSp(8^*|4)$ representations. The best-protected sector consists of half-BPS operators. In the $A_{N-1}$ theory, single-particle supergravity modes on $S^4$ map to protected single-trace-like operators with $SO(5)_R$ quantum numbers.

A schematic tower is

$$\mathcal O_k \quad\longleftrightarrow\quad \text{KK harmonic of degree }k\text{ on }S^4, \qquad \Delta=2k, \qquad k=2,3,\ldots.$$

The $k=2$ multiplet contains the stress tensor, the $SO(5)_R$ current, and other protected operators. In the free center-of-mass tensor multiplet one also has $k=1$ fields, but the interacting $A_{N-1}$ sector begins differently. As always in holography, one should separate the decoupled center-of-mass sector from the interacting large-$N$ theory.

The most important universal dictionary entries are:

CFT$_6$ operator/data	Bulk dual
stress tensor $T_{\alpha\beta}$	AdS$_7$ graviton
$SO(5)_R$ current	gauge fields from $S^4$ isometries
scalar half-BPS operators	Kaluza-Klein scalar modes on $S^4$
surface operators	M2-branes ending on boundary surfaces
codimension defects	probe M2/M5 branes with AdS submanifolds
thermal state	AdS$_7$ black brane
$C_T$, anomaly coefficients	$L^5/G_7\sim N^3$

This dictionary is less elementary-looking than the D3-brane one because the boundary theory lacks a simple Lagrangian. Nevertheless, the protected spectrum, anomalies, correlator normalizations, and thermodynamics are sharply constrained.

Self-dual strings and surface operators

A six-dimensional two-form gauge potential naturally couples to a two-dimensional surface, not to a one-dimensional line. For an abelian tensor multiplet, the analog of a Wilson loop is a Wilson surface:

$$W(\Sigma) \sim \exp\left(i\int_{\Sigma}B\right),$$

where $\Sigma$ is a two-dimensional surface in spacetime. In the interacting theory this formula is only a mnemonic, but the geometry is real: M2-branes can end on M5-branes. The boundary of the M2-brane is a string living on the M5-brane worldvolume. These are the self-dual strings of the $(2,0)$ theory.

In the holographic description, a surface operator in the boundary CFT is represented semiclassically by an M2-brane whose worldvolume ends on the prescribed boundary surface:

$$\langle W(\Sigma)\rangle \sim \exp\left(-T_{\mathrm{M2}}\,\mathrm{Vol}_{\mathrm{ren}}(M_3)\right), \qquad \partial M_3=\Sigma.$$

Here

$$T_{\mathrm{M2}}=\frac{1}{(2\pi)^2\ell_p^3}.$$

Since

$$T_{\mathrm{M2}}L^3\sim N,$$

surface-operator exponents often scale like $N$ at leading semiclassical order. Wrapped M5-branes give other extended observables whose scaling can be larger. The detailed classification of defects in the $(2,0)$ theory is a rich subject, but the basic lesson is simple:

$$\boxed{ \text{In six dimensions, the natural nonlocal probes are surfaces and defects, not only Wilson lines.} }$$

This is one reason the $(2,0)$ theory looks exotic from a four-dimensional gauge-theory viewpoint.

Compactification as a definition machine

The $(2,0)$ theory becomes more familiar after compactification. In fact, many of our best definitions of its properties come from asking what it becomes on lower-dimensional spaces.

Compactification on $S^1$: five-dimensional maximally supersymmetric Yang-Mills

Compactify one spatial direction on a circle of radius $R_6$:

$$\text{$(2,0)$ theory on }S^1_{R_6} \quad\leadsto\quad \text{5d maximally supersymmetric Yang-Mills at low energy.}$$

The five-dimensional gauge coupling has dimension of length:

$$[g_5^2]=\text{length}.$$

The relation is convention-dependent, but schematically

$$\boxed{ g_5^2\sim R_6. }$$

In a common normalization,

$$g_5^2=8\pi^2R_6.$$

Five-dimensional Yang-Mills is perturbatively nonrenormalizable, so it cannot be a complete UV theory by itself. The $(2,0)$ theory is its proposed UV completion. A key check is the instanton particle. In five dimensions, an instanton in the spatial $\mathbb R^4$ becomes a particle with mass

$$M_{\mathrm{inst}}=\frac{8\pi^2}{g_5^2}.$$

Using $g_5^2=8\pi^2R_6$, this is

$$M_{\mathrm{inst}}=\frac{1}{R_6},$$

which is exactly the Kaluza-Klein scale of momentum around the compact circle. Thus the tower of instanton particles knows about the sixth dimension.

Compactification on $T^2$: four-dimensional $\mathcal N=4$ SYM and S-duality

Compactify the $(2,0)$ theory on a two-torus:

$$\text{$(2,0)$ theory on }T^2 \quad\leadsto\quad \text{4d }\mathcal N=4\text{ SYM}.$$

The complex structure of the torus becomes the complexified Yang-Mills coupling:

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

The modular group of the torus,

$$SL(2,\mathbb Z),$$

acts on $\tau_{T^2}$. From the four-dimensional viewpoint this is the S-duality group of $\mathcal N=4$ SYM. This is one of the most elegant explanations of why electric-magnetic duality in four dimensions should be exact: it is geometry in six dimensions.

Compactification on a Riemann surface: class-$\mathcal S$ theories

Compactify the $(2,0)$ theory on a Riemann surface $C$ with a partial topological twist:

$$\text{$(2,0)$ theory of type }\mathfrak g\text{ on }C \quad\leadsto\quad \text{4d }\mathcal N=2\text{ class-}\mathcal S\text{ theory}.$$

The complex structure moduli of $C$ become exactly marginal couplings of the four-dimensional theory. Degeneration limits of $C$ correspond to weakly coupled gauge-theory frames. Punctures on $C$ correspond to codimension-two defects of the six-dimensional theory and produce flavor symmetries in four dimensions.

This is far beyond a historical curiosity. It means that many dualities between four-dimensional gauge theories are shadows of the same six-dimensional parent theory viewed through different pair-of-pants decompositions of $C$.

What makes the $(2,0)$ theory hard?

The theory is difficult for several intertwined reasons.

First, six-dimensional conformal field theories are highly constrained. A Yang-Mills coupling in six dimensions has negative mass dimension:

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

So ordinary six-dimensional Yang-Mills is not a fundamental conformal theory. The $(2,0)$ theory is not obtained by tuning a six-dimensional gauge coupling to a fixed point in the usual perturbative way.

Second, the basic tensor multiplet contains a chiral two-form. Even abelian chiral $p$-form theories have subtleties involving self-duality, partition functions, and anomalies. The interacting nonabelian version is much more subtle.

Third, the theory contains string-like excitations. On the Coulomb branch, separated M5-branes allow M2-branes to stretch between them. Their boundaries are strings in the M5 worldvolume, with tension proportional to the brane separation. At the conformal point, where the branes coincide, these strings become tensionless.

Fourth, many observables are naturally extended rather than local. Surface operators, defects, compactification data, and anomaly polynomials are often better handles than elementary fields.

These difficulties should not be mistaken for vagueness. The theory has sharp protected data, a precise holographic dual at large $N$, and numerous compactification checks.

The anomaly viewpoint

In even-dimensional CFTs, the stress-tensor trace on a curved background has an anomaly:

$$\langle T^\alpha_{\ \alpha}\rangle = \sum_i c_i I_i - a E_6 + \nabla_\alpha J^\alpha.$$

Here $E_6$ is the six-dimensional Euler density, the $I_i$ are Weyl invariants, and the last term is scheme-dependent. In the $(2,0)$ theory, supersymmetry relates much of this anomaly data. Holographically, the anomaly is extracted by evaluating the renormalized on-shell action of seven-dimensional asymptotically AdS gravity.

The crucial large-$N$ scaling is

$$\boxed{ a_{6d}\sim C_T\sim N^3. }$$

More refined formulas include subleading terms and distinguish the interacting $A_{N-1}$ theory from the decoupled center-of-mass tensor multiplet. For the purposes of the present course, the main lesson is that anomaly coefficients are the six-dimensional analogs of central charges. They measure the normalization of stress-tensor correlators and therefore the effective inverse Newton constant of the AdS$_7$ dual.

This is the same logic used throughout AdS/CFT:

$$\boxed{ \text{CFT stress-tensor normalization} \quad\longleftrightarrow\quad \text{bulk gravitational coupling}^{-1}. }$$

For M5-branes, this normalization is order $N^3$.

Relation to other AdS$_7$ duals

The maximally supersymmetric $\mathrm{AdS}_7\times S^4$ solution is the cleanest AdS$_7$/CFT$_6$ example, but it is not the only context in which six-dimensional SCFTs appear in holography. Less supersymmetric six-dimensional $(1,0)$ SCFTs can also have AdS$_7$ duals in massive type IIA or M-theory constructions. Those theories involve richer flavor symmetries, brane intersections, orientifolds, and anomaly-matching tests.

However, the maximally supersymmetric $(2,0)$ theory remains the basic reference point because:

• the geometry is simple and explicit;
• the symmetry is maximal;
• the $N^3$ scaling is clean;
• compactifications connect directly to many lower-dimensional dualities;
• the large-$N$ supergravity regime is controlled by one parameter, $N$.

The broader AdS$_7$/CFT$_6$ landscape is important for research, but it is best understood after the M5-brane example is mastered.

Common mistakes

Mistake 1: Calling the six-dimensional theory “Yang-Mills”

The $(2,0)$ theory is not six-dimensional Yang-Mills. Five-dimensional maximally supersymmetric Yang-Mills appears after compactification on $S^1$ and is a low-energy description below the Kaluza-Klein scale.

Mistake 2: Forgetting the free center-of-mass tensor multiplet

The worldvolume theory of $N$ M5-branes contains a free center-of-mass tensor multiplet plus the interacting $A_{N-1}$ theory. Holographic large-$N$ statements usually refer to the interacting sector, where the $N^3$ scaling dominates.

Mistake 3: Treating $N^3$ as ordinary matrix counting

The $N^3$ scaling is not explained by adjoint fields with two color indices. It is a gravitational result from $L^5/G_7\sim N^3$ and is one of the reasons M5-branes are conceptually deeper than ordinary large-$N$ gauge theories.

Mistake 4: Looking for a tunable ‘t Hooft coupling

There is no analog of $\lambda=g_{\mathrm{YM}}^2N$ in the uncompactified $(2,0)$ theory. The large-$N$ classical bulk limit is controlled by $L/\ell_p\sim N^{1/3}$.

Mistake 5: Thinking compactification loses information

Compactification is not merely a way to approximate the theory. It is one of the most powerful ways to define and test the theory. The geometry of the compactification manifold becomes duality data of the lower-dimensional theory.

Exercises

Exercise 1: The M5 near-horizon geometry

Start from the extremal M5-brane metric

$$ds_{11}^2 = H^{-1/3}dx_{1,5}^2 + H^{2/3}\left(dr^2+r^2d\Omega_4^2\right), \qquad H=1+\frac{R^3}{r^3}.$$

Show that the near-horizon geometry is $\mathrm{AdS}_7\times S^4$ and determine the ratio of the two radii.

Solution

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

$$H\simeq \frac{R^3}{r^3}.$$

Therefore

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

The metric becomes

$$ds_{11}^2 = \frac{r}{R}dx_{1,5}^2 + \frac{R^2}{r^2}dr^2 + R^2d\Omega_4^2.$$

Define

$$r=\frac{4R^3}{z^2}.$$

Then

$$\frac{r}{R}dx_{1,5}^2+\frac{R^2}{r^2}dr^2 = \frac{4R^2}{z^2}\left(dx_{1,5}^2+dz^2\right).$$

Thus

$$ds_{11}^2 = (2R)^2\frac{dx_{1,5}^2+dz^2}{z^2} + R^2d\Omega_4^2.$$

So

$$L_{\mathrm{AdS}_7}=2R, \qquad L_{S^4}=R.$$

Exercise 2: Why $L^5/G_7\sim N^3$

Assume

$$L^3=8\pi N\ell_p^3, \qquad G_{11}=16\pi^7\ell_p^9, \qquad \mathrm{Vol}(S^4_{L/2})=\frac{8\pi^2}{3}\left(\frac{L}{2}\right)^4.$$

Show that

$$\frac{L^5}{G_7}\propto N^3.$$

Solution

Dimensional reduction gives

$$\frac{1}{G_7}=\frac{\mathrm{Vol}(S^4_{L/2})}{G_{11}}.$$

Thus

$$\frac{L^5}{G_7} = L^5\frac{\mathrm{Vol}(S^4_{L/2})}{G_{11}} = L^5\frac{1}{G_{11}}\frac{8\pi^2}{3}\left(\frac{L}{2}\right)^4.$$

This is proportional to

$$\frac{L^9}{\ell_p^9}.$$

Since

$$L^3\propto N\ell_p^3,$$

we have

$$L^9\propto N^3\ell_p^9.$$

Therefore

$$\frac{L^5}{G_7}\propto N^3.$$

Keeping the stated conventions gives

$$\frac{L^5}{G_7}=\frac{16}{3\pi^2}N^3.$$

Exercise 3: Entropy density of the M5 plasma

For the planar AdS$_7$ black brane,

$$ds_7^2 = \frac{L^2}{z^2}\left[-f(z)dt^2+d\vec x_5^{\,2}+\frac{dz^2}{f(z)}\right], \qquad f(z)=1-\left(\frac{z}{z_h}\right)^6,$$

show that the entropy density scales as

$$s\sim N^3T^5.$$

Solution

The Hawking temperature of the AdS$_{d+1}$ black brane is

$$T=\frac{d}{4\pi z_h}.$$

Here $d=6$, so

$$T=\frac{3}{2\pi z_h}, \qquad \frac{1}{z_h}=\frac{2\pi T}{3}.$$

The horizon area density is

$$\left(\frac{L}{z_h}\right)^5.$$

Therefore

$$s = \frac{1}{4G_7}\left(\frac{L}{z_h}\right)^5 = \frac{L^5}{4G_7}\left(\frac{2\pi T}{3}\right)^5.$$

Since $L^5/G_7\sim N^3$,

$$s\sim N^3T^5.$$

With the conventions of Exercise 2,

$$s=\frac{128\pi^3}{729}N^3T^5.$$

Exercise 4: Instantons as Kaluza-Klein modes

Compactify the $(2,0)$ theory on a circle of radius $R_6$. Suppose the low-energy five-dimensional Yang-Mills coupling is normalized as

$$g_5^2=8\pi^2R_6.$$

Show that the five-dimensional instanton-particle mass equals the Kaluza-Klein mass of one unit of momentum around the circle.

Solution

In five-dimensional maximally supersymmetric Yang-Mills, an instanton in the four spatial directions is a particle. Its mass is

$$M_{\mathrm{inst}}=\frac{8\pi^2}{g_5^2}.$$

Using

$$g_5^2=8\pi^2R_6,$$

we find

$$M_{\mathrm{inst}}=\frac{1}{R_6}.$$

A Kaluza-Klein excitation with one unit of momentum around a circle of radius $R_6$ also has mass

$$M_{\mathrm{KK}}=\frac{1}{R_6}.$$

Thus instanton number in five dimensions is identified with momentum around the compact sixth dimension.

Exercise 5: Why S-duality becomes geometry

Explain why compactifying the $(2,0)$ theory on a two-torus gives a natural origin for the $SL(2,\mathbb Z)$ duality of four-dimensional $\mathcal N=4$ SYM.

Solution

The two-torus has a mapping class group

$$SL(2,\mathbb Z),$$

which acts on its complex structure $\tau_{T^2}$. Compactification of the $(2,0)$ theory on $T^2$ gives four-dimensional $\mathcal N=4$ SYM, and the torus complex structure maps to the complexified gauge coupling,

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

Therefore the geometric modular transformations of $T^2$ act as duality transformations on $\tau_{\mathrm{YM}}$. The electric-magnetic S-duality of four-dimensional $\mathcal N=4$ SYM is reinterpreted as ordinary geometry of the compactification torus.

Further reading

• J. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity. The original AdS/CFT paper, including the M2- and M5-brane examples.
• O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, Large N Field Theories, String Theory and Gravity. The standard review, with a useful discussion of AdS$_7$/CFT$_6$.
• M. Henningson and K. Skenderis, The Holographic Weyl Anomaly. The classic holographic computation showing the $N^3$ scaling of the six-dimensional anomaly.
• E. Witten, Some Comments on String Dynamics. Early perspective on fivebranes, tensionless strings, and the six-dimensional theory.
• D. Gaiotto, N=2 Dualities. The class-$\mathcal S$ viewpoint from compactifying the six-dimensional $(2,0)$ theory on Riemann surfaces.
• D. Gaiotto, G. Moore, and Y. Tachikawa, On 6d N=(2,0) Theory Compactified on a Riemann Surface with Finite Area. A more detailed study of compactification subtleties.