AdS3/CFT2 and Brown-Henneaux

The main idea

AdS$_3$/CFT$_2$ is the most symmetry-rich corner of holography. In dimensions $d+1\neq 3$, the isometry group of AdS$_{d+1}$ matches the finite-dimensional conformal group of the $d$-dimensional boundary theory. In AdS$_3$, the boundary is two-dimensional, and local conformal transformations form an infinite-dimensional algebra. The remarkable Brown-Henneaux result is that this infinite symmetry is not merely a boundary-theory wish: it is already visible in classical three-dimensional gravity with negative cosmological constant.

For Einstein gravity on asymptotically AdS$_3$ spacetimes,

$$S = \frac{1}{16\pi G_3} \int_{\mathcal M} d^3x\sqrt{-g}\left(R+\frac{2}{L^2}\right) +S_{\mathrm{bdy}},$$

the asymptotic symmetry algebra is

$$\boxed{ \mathrm{Virasoro}_L\times \mathrm{Virasoro}_R }$$

with equal classical central charges

$$\boxed{ c_L=c_R=\frac{3L}{2G_3}. }$$

This formula is one of the sharpest bridges between geometry and CFT data in all of AdS/CFT. The large-central-charge limit of the CFT$_2$ is the classical-gravity limit of the bulk:

$$\frac{L}{G_3}\gg 1 \qquad\Longleftrightarrow\qquad c\gg 1.$$

The page has two goals. First, it explains why AdS$_3$ gravity has an infinite-dimensional boundary symmetry even though pure three-dimensional Einstein gravity has no propagating gravitons. Second, it explains how the BTZ black-hole entropy follows from the Cardy formula using the Brown-Henneaux central charge.

Brown-Henneaux asymptotic symmetry. Global AdS$_3$ has finite isometry group $SO(2,2)\simeq SL(2,\mathbb R)_L\times SL(2,\mathbb R)_R$, but the allowed asymptotic diffeomorphisms preserving AdS$_3$ boundary conditions enlarge this to two Virasoro algebras. The central charge $c=3L/(2G_3)$ is the basic CFT$_2$ measure of the number of degrees of freedom.

The slogan is therefore:

$$\boxed{ \text{AdS}_3\text{ gravity is locally simple but asymptotically rich.} }$$

That one sentence prevents many misunderstandings. Pure Einstein gravity in three dimensions has no local graviton waves, but it still has black holes, boundary gravitons, nontrivial topology, Virasoro charges, and a highly constrained dual CFT$_2$ interpretation.

Why three dimensions are special

In three bulk dimensions the Weyl tensor vanishes identically. The Riemann tensor is completely determined by the Ricci tensor:

$$R_{abcd} = g_{ac}R_{bd}-g_{ad}R_{bc}-g_{bc}R_{ad}+g_{bd}R_{ac} -\frac{R}{2}\left(g_{ac}g_{bd}-g_{ad}g_{bc}\right).$$

The vacuum Einstein equation with negative cosmological constant is

$$R_{ab}=-\frac{2}{L^2}g_{ab}, \qquad R=-\frac{6}{L^2}.$$

Substituting into the identity gives

$$R_{abcd} =-\frac{1}{L^2}\left(g_{ac}g_{bd}-g_{ad}g_{bc}\right).$$

Thus every smooth vacuum solution is locally AdS$_3$. There are no local curvature degrees of freedom in pure three-dimensional Einstein gravity. The usual graviton polarization count in $D$ spacetime dimensions is

$$N_{\mathrm{graviton}} =\frac{D(D-3)}{2},$$

so $D=3$ gives zero propagating graviton polarizations.

This does not mean the theory is empty. The physical content shifts from local waves to global and boundary data:

Bulk feature	Why it matters in AdS$_3$/CFT$_2$
Boundary gravitons	Large diffeomorphisms are physical because they act nontrivially at infinity.
BTZ black holes	Black holes are global quotients of AdS$_3$, not local curvature lumps.
Virasoro charges	The boundary stress tensor organizes states into left/right conformal families.
Topology and saddles	Different Euclidean fillings contribute to the gravitational path integral.
Chern-Simons structure	Pure AdS$_3$ gravity can be rewritten as an $SL(2,\mathbb R)\times SL(2,\mathbb R)$ Chern-Simons theory.

A useful analogy is electromagnetism in two spatial dimensions with boundaries. If there are no local waves in a sector, there can still be boundary charges, holonomies, and edge modes. In AdS$_3$ gravity, this boundary structure is not an optional decoration. It is the mechanism by which the theory knows about a two-dimensional CFT.

Global AdS$_3$ and the boundary cylinder

Global AdS$_3$ can be written as

$$ds^2 =-\left(1+\frac{r^2}{L^2}\right)dt^2 +\frac{dr^2}{1+r^2/L^2} +r^2d\phi^2, \qquad \phi\sim \phi+2\pi.$$

At large $r$,

$$ds^2 \sim \frac{r^2}{L^2}\left(-dt^2+L^2d\phi^2\right) +\frac{L^2}{r^2}dr^2.$$

After the usual conformal rescaling, the boundary metric is the cylinder

$$ds^2_{\partial} =-dt^2+L^2d\phi^2.$$

It is convenient to introduce dimensionless light-cone coordinates

$$x^+=\frac{t}{L}+\phi, \qquad x^-=\frac{t}{L}-\phi.$$

The global isometry group is

$$SO(2,2) \simeq \frac{SL(2,\mathbb R)_L\times SL(2,\mathbb R)_R}{\mathbb Z_2}.$$

This is precisely the global conformal group of a two-dimensional CFT on the cylinder. The Brown-Henneaux result says that once one allows diffeomorphisms that preserve the asymptotic AdS$_3$ falloffs, the finite-dimensional group above is enhanced to the full local conformal group:

$$SL(2,\mathbb R)_L\times SL(2,\mathbb R)_R \quad\longrightarrow\quad \mathrm{Vir}_L\times \mathrm{Vir}_R.$$

The three generators $L_{-1},L_0,L_1$ form the global $SL(2,\mathbb R)_L$, and similarly $\bar L_{-1},\bar L_0,\bar L_1$ form $SL(2,\mathbb R)_R$. All other Virasoro generators are asymptotic symmetries rather than exact Killing vectors of global AdS$_3$.

Brown-Henneaux boundary conditions

An asymptotically AdS$_3$ metric should approach global AdS$_3$ at large $r$, but it need not equal global AdS$_3$ exactly. Brown and Henneaux imposed falloff conditions of the schematic form

$$\begin{aligned} g_{tt}&=-\frac{r^2}{L^2}+O(1), & g_{t\phi}&=O(1), & g_{\phi\phi}&=r^2+O(1),\\ g_{rr}&=\frac{L^2}{r^2}+O(r^{-4}), & g_{rt}&=O(r^{-3}), & g_{r\phi}&=O(r^{-3}). \end{aligned}$$

These boundary conditions are strong enough to define finite conserved charges and weak enough to include physically important geometries such as BTZ black holes and boundary gravitons.

The crucial question is: which diffeomorphisms preserve these falloffs? Near the boundary, the answer is controlled by two arbitrary functions,

$$\epsilon^+(x^+), \qquad \epsilon^-(x^-).$$

Thus the asymptotic transformations act chirally:

$$x^+\to x^++\epsilon^+(x^+), \qquad x^-\to x^-+\epsilon^-(x^-).$$

Expanding in Fourier modes,

$$\epsilon^+(x^+)=\sum_n \epsilon_n e^{inx^+}, \qquad \epsilon^-(x^-)=\sum_n \bar\epsilon_n e^{inx^-},$$

one obtains generators $L_n$ and $\bar L_n$. Classically their Poisson brackets have a central extension. Quantum mechanically this becomes the Virasoro algebra

$$\boxed{ [L_m,L_n] =(m-n)L_{m+n} +\frac{c}{12}m(m^2-1)\delta_{m+n,0}, }$$ $$\boxed{ [\bar L_m,\bar L_n] =(m-n)\bar L_{m+n} +\frac{c}{12}m(m^2-1)\delta_{m+n,0}, \qquad [L_m,\bar L_n]=0. }$$

The central term is written with $m(m^2-1)$ because this basis makes the global subalgebra $m=0,\pm 1$ have no central extension. In another common basis one sees $m^3\delta_{m+n,0}$; the difference is a shift of $L_0$ by the vacuum Casimir energy.

The central charge is

$$\boxed{ c=\frac{3L}{2G_3}. }$$

This number is not fitted from a black-hole entropy calculation. It is determined by the algebra of canonical boundary charges. That is why the later Cardy/BTZ agreement is so powerful: the same $c$ that appears in the asymptotic symmetry algebra controls the high-energy density of states.

Boundary gravitons

A small diffeomorphism that vanishes sufficiently fast at the boundary is a gauge redundancy. A large diffeomorphism that changes the boundary charges is not pure gauge. Acting with such a transformation on global AdS$_3$ gives a physically distinct state called a boundary graviton.

In CFT language, the vacuum $|0\rangle$ has descendants

$$L_{-n_1}\cdots L_{-n_k}\bar L_{-m_1}\cdots \bar L_{-m_\ell}|0\rangle, \qquad n_i,m_j\ge 2.$$

The $n=1$ modes are global conformal transformations, and the $n=0$ mode measures energy/spin. The $n\ge 2$ descendants are the CFT image of boundary gravitons.

This is a useful place to distinguish three ideas that are easily conflated:

Statement	Correct meaning
“No local gravitons in AdS$_3$”	No propagating bulk spin-2 wave packets in pure Einstein gravity.
“There are boundary gravitons”	Large diffeomorphisms create physical Virasoro descendants.
“The theory is a CFT$_2$”	The Hilbert space and observables organize under two Virasoro algebras with Brown-Henneaux central charge.

A theory can have no local bulk gravitons and still have an infinite-dimensional Hilbert-space structure at the boundary. AdS$_3$ is the canonical example.

The stress tensor viewpoint

The Brown-Henneaux result can also be understood through the holographic stress tensor. In Fefferman-Graham gauge,

$$ds^2 =L^2d\rho^2+g_{ij}(\rho,x)dx^idx^j,$$

an asymptotically AdS$_3$ metric has the expansion

$$g_{ij}(\rho,x) =e^{2\rho}g^{(0)}_{ij}(x) +g^{(2)}_{ij}(x) +e^{-2\rho}g^{(4)}_{ij}(x).$$

In three bulk dimensions the expansion terminates locally for pure Einstein gravity. The coefficient $g^{(0)}_{ij}$ is the boundary metric source, and $g^{(2)}_{ij}$ determines the expectation value of the CFT stress tensor. Up to convention-dependent signs associated with Lorentzian versus Euclidean signature,

$$\langle T_{ij}\rangle =\frac{L}{8\pi G_3} \left( g^{(2)}_{ij}-g^{(0)}_{ij}\operatorname{tr}g^{(2)} \right).$$

The trace anomaly of a two-dimensional CFT is

$$\langle T^i{}_i\rangle =-\frac{c}{24\pi}R[g^{(0)}],$$

again up to sign conventions for the curvature. Holographic renormalization gives the same anomaly with

$$c=\frac{3L}{2G_3}.$$

This stress-tensor derivation is conceptually useful because it connects the central charge to a familiar CFT observable: the normalization of the stress-tensor two-point function and the Weyl anomaly. In AdS$_3$/CFT$_2$, the Brown-Henneaux central charge is simultaneously

$$\boxed{ \text{asymptotic-symmetry central charge} = \text{CFT Weyl-anomaly coefficient} = \text{stress-tensor normalization.} }$$

Stress tensor transformations and the Schwarzian

A two-dimensional CFT stress tensor does not transform as an ordinary tensor under a conformal map. On the complex plane, if $z=f(w)$, then

$$T(w) =\left(\frac{df}{dw}\right)^2 T(f(w)) +\frac{c}{12}\{f,w\},$$

where

$$\{f,w\} =\frac{f'''(w)}{f'(w)} -\frac{3}{2}\left(\frac{f''(w)}{f'(w)}\right)^2$$

is the Schwarzian derivative. The Schwarzian term is the local signature of the central charge.

The same structure appears geometrically in AdS$_3$. A large diffeomorphism preserving Brown-Henneaux boundary conditions changes the subleading metric data, hence the boundary stress tensor, by a transformation law containing a Schwarzian. In this sense, boundary gravitons are the geometric realization of Virasoro descendants.

The vacuum energy on the cylinder is another elementary example. Mapping the plane to the cylinder produces

$$E_{\mathrm{vac}} =-\frac{c}{12L}$$

for a CFT on a circle of radius $L$, where both left and right movers contribute. On the gravity side, global AdS$_3$ has

$$M_{\mathrm{AdS}}=-\frac{1}{8G_3}.$$

Using $c=3L/(2G_3)$ gives

$$-\frac{c}{12L}=-\frac{1}{8G_3},$$

so the cylinder Casimir energy matches the mass of global AdS$_3$.

BTZ black holes

The BTZ black hole is the basic thermal object in AdS$_3$/CFT$_2$. In coordinates adapted to stationarity and rotation,

$$ds^2 =-N^2dt^2+\frac{dr^2}{N^2} +r^2\left(d\phi+N^\phi dt\right)^2,$$

with

$$N^2 =-8G_3M+\frac{r^2}{L^2}+\frac{16G_3^2J^2}{r^2}, \qquad N^\phi=-\frac{4G_3J}{r^2}.$$

Equivalently, in terms of outer and inner horizon radii $r_+$ and $r_-$,

$$M=\frac{r_+^2+r_-^2}{8G_3L^2}, \qquad J=\frac{r_+r_-}{4G_3L}.$$

The Hawking temperature and angular velocity are

$$T_H =\frac{r_+^2-r_-^2}{2\pi L^2r_+}, \qquad \Omega_H=\frac{r_-}{Lr_+}.$$

The entropy is the horizon “area” divided by $4G_3$. In three bulk dimensions the horizon area is a circumference:

$$\boxed{ S_{\mathrm{BH}} =\frac{2\pi r_+}{4G_3} =\frac{\pi r_+}{2G_3}. }$$

The geometry is locally AdS$_3$. The black hole arises from a global quotient, not from a local curvature singularity in the same way as a higher-dimensional Schwarzschild black hole. Nevertheless it has a genuine horizon, temperature, entropy, and causal structure. This is one reason BTZ black holes are such clean laboratories for quantum gravity.

BTZ as a CFT$_2$ thermal state

A rotating BTZ black hole corresponds to a CFT$_2$ state with independent left- and right-moving temperatures

$$T_L=\frac{r_+-r_-}{2\pi L^2}, \qquad T_R=\frac{r_++r_-}{2\pi L^2},$$

up to the convention for which chirality is called left or right. These combine into the Hawking temperature through

$$\frac{2}{T_H}=\frac{1}{T_L}+\frac{1}{T_R}.$$

The CFT Hamiltonian and angular momentum on a circle of radius $L$ are

$$H=\frac{1}{L}\left(L_0+\bar L_0-\frac{c}{12}\right), \qquad J_{\mathrm{CFT}}=L_0-\bar L_0.$$

Thus the shifted conformal weights are related to the bulk mass and spin by

$$L_0-\frac{c}{24}=\frac{ML+J}{2}, \qquad \bar L_0-\frac{c}{24}=\frac{ML-J}{2}.$$

Global AdS$_3$ has $M=-1/(8G_3)$ and $J=0$, so

$$L_0=\bar L_0=0.$$

The massless BTZ geometry has $M=J=0$, so

$$L_0=\bar L_0=\frac{c}{24}.$$

This is the familiar distinction between the vacuum on the plane and the zero-energy threshold above the cylinder vacuum.

Cardy’s formula and BTZ entropy

For a unitary modular-invariant two-dimensional CFT, the asymptotic density of states at large left/right weights is controlled by Cardy’s formula:

$$S_{\mathrm{Cardy}} =2\pi\sqrt{\frac{c_L}{6}\left(L_0-\frac{c_L}{24}\right)} +2\pi\sqrt{\frac{c_R}{6}\left(\bar L_0-\frac{c_R}{24}\right)}.$$

For pure Einstein gravity, $c_L=c_R=c=3L/(2G_3)$. Using

$$L_0-\frac{c}{24}=\frac{ML+J}{2}, \qquad \bar L_0-\frac{c}{24}=\frac{ML-J}{2},$$

and the BTZ relations

$$ML+J=\frac{(r_++r_-)^2}{8G_3L}, \qquad ML-J=\frac{(r_+-r_-)^2}{8G_3L},$$

Cardy’s formula gives

$$\begin{aligned} S_{\mathrm{Cardy}} &=2\pi\sqrt{ \frac{1}{6}\frac{3L}{2G_3}\frac{(r_++r_-)^2}{16G_3L} } +2\pi\sqrt{ \frac{1}{6}\frac{3L}{2G_3}\frac{(r_+-r_-)^2}{16G_3L} }\\ &=\frac{\pi}{4G_3}\left[(r_++r_-)+(r_+-r_-)\right]\\ &=\frac{\pi r_+}{2G_3}. \end{aligned}$$

Therefore

$$\boxed{ S_{\mathrm{Cardy}}=S_{\mathrm{BH}}. }$$

This is one of the cleanest microscopic entropy computations in holography. The calculation uses only asymptotic symmetry and modular invariance, not the detailed stringy microstates of a particular compactification. In string-theoretic AdS$_3$ backgrounds, such as the D1-D5 system, the same logic is supplemented by an explicit CFT$_2$ with central charge proportional to the brane charges.

What exactly is being counted?

The Cardy formula counts the asymptotic growth of CFT$_2$ states at high conformal weights. The BTZ black hole is a coarse-grained bulk description of many such states. The geometry is not usually one exact microstate. It is a thermodynamic saddle representing an ensemble, or a coarse-grained description of heavy states.

This distinction matters. The equality

$$S_{\mathrm{Cardy}}=\frac{\operatorname{Area}}{4G_3}$$

does not mean that classical geometry has identified every individual microstate. It means that the number of states implied by the dual CFT, constrained by symmetry and modular invariance, has the same leading entropy as the black-hole horizon.

A useful hierarchy is:

Object	CFT$_2$ description	Bulk description
Vacuum	$L_0=\bar L_0=0$ on the plane	Global AdS$_3$
Boundary graviton	Virasoro descendant of vacuum or another primary	Large diffeomorphism of an AdS$_3$ geometry
Light primary	Low-dimension primary state	Particle or conical defect, depending on dimensions
Heavy primary	$h,\bar h\sim c$	Backreacted geometry; often BTZ-like above threshold
Thermal ensemble	Torus partition function	Euclidean BTZ or thermal AdS saddle
BTZ black hole	High-energy density of states	Locally AdS$_3$ quotient with horizon

The phrase “heavy state” often means $h,\bar h$ scale like $c$ in the large-$c$ limit. Such states can create classical backreaction. The precise bulk interpretation depends on the spectrum and on whether the state is below or above the black-hole threshold.

Pure gravity, string theory, and the dual CFT

It is tempting to say: “pure AdS$_3$ Einstein gravity is dual to a CFT$_2$ with $c=3L/(2G_3)$.” This is a useful semiclassical slogan, but it hides a deep problem. A complete quantum theory must have a consistent Hilbert space, a modular-invariant torus partition function, positive degeneracies, and sensible correlation functions. The Brown-Henneaux analysis tells us the asymptotic symmetry algebra and central charge. It does not by itself construct the full dual CFT.

This is why there are several levels of AdS$_3$/CFT$_2$:

Level	What is known
Classical Einstein-AdS$_3$	Brown-Henneaux Virasoro symmetry, BTZ black holes, boundary gravitons.
Semiclassical pure gravity	Powerful but subtle; the full nonperturbative CFT dual is not automatically guaranteed.
Chern-Simons formulation	Rewrites gravity as $SL(2,\mathbb R)_L\times SL(2,\mathbb R)_R$ Chern-Simons theory with level $k=L/(4G_3)$.
Stringy AdS$_3$ backgrounds	Often have explicit candidate CFT$_2$ duals, such as the D1-D5 system.
Higher-spin AdS$_3$	Enlarged asymptotic symmetry algebras such as $\mathcal W$-algebras.

In the Chern-Simons formulation,

$$S_{\mathrm{grav}} =S_{\mathrm{CS}}[A]-S_{\mathrm{CS}}[\bar A],$$

with gauge group $SL(2,\mathbb R)_L\times SL(2,\mathbb R)_R$ and level

$$k=\frac{L}{4G_3}.$$

The Brown-Henneaux central charge is then

$$c=6k.$$

This reformulation is extremely useful for boundary charges, higher-spin generalizations, and Euclidean saddles. But it is not a magic replacement for the full quantum theory: boundary conditions, allowed holonomies, modular invariance, and the spectrum all matter.

The D1-D5 example: a stringy AdS$_3$/CFT$_2$

The most important top-down AdS$_3$/CFT$_2$ example comes from the D1-D5 system. Type IIB string theory on backgrounds of the form

$$\mathrm{AdS}_3\times S^3\times \mathcal M_4, \qquad \mathcal M_4=T^4 \text{ or } K3,$$

is dual to a two-dimensional CFT with supersymmetry. In a common regime, the central charge is

$$c=6Q_1Q_5,$$

where $Q_1$ and $Q_5$ are D1- and D5-brane charges. The Brown-Henneaux formula agrees with this after reducing the ten-dimensional geometry to three dimensions:

$$\frac{3L}{2G_3}=6Q_1Q_5.$$

The D1-D5 system is not merely a toy model. It is the arena in which many ideas about black-hole microstates, elliptic genera, symmetric-product orbifolds, AdS$_3$ strings, and precision holography have been developed. But the point of the present page is broader: the Virasoro structure of AdS$_3$ gravity is universal, while each string compactification supplies a particular microscopic CFT.

Thermal AdS, Euclidean BTZ, and modular transformations

A two-dimensional CFT at finite temperature lives on a Euclidean torus. The two cycles of the torus can be exchanged by a modular transformation. In the bulk, this exchange corresponds to changing which boundary cycle becomes contractible in the Euclidean filling.

Two important saddles are:

Boundary torus filling	Contractible cycle	CFT interpretation
Thermal AdS$_3$	Spatial circle	Low-temperature vacuum-dominated phase
Euclidean BTZ	Thermal circle	High-temperature black-hole phase

This is the three-dimensional version of the Hawking-Page story, but now it is tightly linked to modular invariance of the CFT$_2$ torus partition function. At high temperature, modular invariance maps the partition function to a low-temperature vacuum-dominated channel, and Cardy’s formula follows from the vacuum contribution.

Schematic CFT logic:

$$Z(\beta) =\operatorname{Tr}\,e^{-\beta H} \quad\xrightarrow{\text{modular }S}\quad Z\left(\frac{4\pi^2L^2}{\beta}\right).$$

At small $\beta$, the transformed channel is dominated by the vacuum energy

$$E_{\mathrm{vac}}=-\frac{c}{12L},$$

so the free energy scales as

$$\log Z(\beta) \sim \frac{\pi^2 c L}{3\beta}.$$

This reproduces the entropy of a nonrotating BTZ black hole. The same logic with angular potential gives the left/right Cardy formula.

Relation to earlier modules

AdS$_3$/CFT$_2$ sharpens many themes from earlier parts of the course:

Earlier concept	AdS$_3$/CFT$_2$ refinement
Central charge $C_T$	Becomes the Virasoro central charge $c=3L/(2G_3)$.
Black-hole entropy	BTZ entropy follows from Cardy’s universal density of states.
Boundary stress tensor	Its transformation law contains the Schwarzian derivative.
Large $N$	Replaced by large $c$, often $c\sim Q_1Q_5$ or $c\sim L/G_3$.
Witten diagrams	Strongly constrained by Virasoro symmetry and conformal blocks.
Entanglement entropy	RT geodesics reproduce the universal CFT$_2$ interval formula.
Thermal physics	Euclidean BTZ and thermal AdS are different torus fillings.
Bulk locality	Particularly subtle because pure gravity has no local gravitons, while stringy AdS$_3$ has rich light spectra.

In higher-dimensional AdS/CFT, the conformal group is finite-dimensional and only constrains correlators up to functions of cross-ratios. In CFT$_2$, Virasoro symmetry gives much stronger constraints. This is why AdS$_3$/CFT$_2$ is simultaneously simpler and more subtle than AdS$_5$/CFT$_4$.

Common mistakes

Mistake 1: “No local gravitons” means “no dynamics”

Pure three-dimensional Einstein gravity has no propagating graviton polarizations, but it still has boundary gravitons, BTZ black holes, nontrivial topology, modular sums, and boundary stress-tensor dynamics. The absence of local gravitons makes the theory special, not empty.

Mistake 2: Brown-Henneaux proves every AdS$_3$ gravity theory is pure Einstein gravity

Brown-Henneaux gives the asymptotic symmetry algebra for a specified set of boundary conditions in Einstein-AdS$_3$ gravity. Stringy AdS$_3$ compactifications usually contain many additional fields, branes, long strings, supersymmetry sectors, and internal excitations. They share the Brown-Henneaux gravitational central charge but have more microscopic structure.

Mistake 3: Cardy’s formula is a low-energy formula

Cardy’s formula controls the asymptotic high-energy density of states under assumptions such as unitarity and modular invariance. Applying it to low-lying states requires extra assumptions and is generally not justified.

Mistake 4: The BTZ black hole is a local curvature singularity

The BTZ geometry is locally AdS$_3$ outside the quotient singularity structure. Its black-hole nature comes from global identifications and causal structure. Curvature invariants are constant away from singular quotient regions.

Mistake 5: The central charge is a convention-dependent normalization

Operator normalizations can vary, but the Virasoro central charge is physical. Once $G_3$ and $L$ are fixed in the gravitational action, Brown-Henneaux fixes $c=3L/(2G_3)$.

Mistake 6: Pure AdS$_3$ gravity automatically has a known CFT dual

The asymptotic symmetry analysis is not the same as constructing a complete, unitary, modular-invariant CFT with the correct spectrum. Pure AdS$_3$ quantum gravity remains subtle. Stringy AdS$_3$ examples are better controlled because the microscopic CFT can often be identified.

Exercises

Exercise 1: No local gravitons in three dimensions

Use the formula for the number of on-shell graviton polarizations in $D$ spacetime dimensions,

$$N_{\mathrm{graviton}}=\frac{D(D-3)}{2},$$

to explain why pure Einstein gravity in AdS$_3$ has no local graviton wave packets. Why does this not contradict the existence of boundary gravitons?

Solution

For $D=3$,

$$N_{\mathrm{graviton}}=\frac{3(3-3)}{2}=0.$$

Thus there are no propagating local spin-2 degrees of freedom in pure three-dimensional Einstein gravity. Equivalently, the Weyl tensor vanishes identically in three dimensions, and the vacuum Einstein equation fixes the local curvature to be that of AdS$_3$.

This does not eliminate boundary gravitons. Boundary gravitons are not local wave packets in the bulk. They are produced by large diffeomorphisms that preserve the asymptotic AdS$_3$ boundary conditions but act nontrivially on the boundary charges. In CFT$_2$ language they are Virasoro descendants.

Exercise 2: The Brown-Henneaux central charge and global AdS energy

Show that the CFT$_2$ vacuum energy on a circle of radius $L$,

$$E_{\mathrm{vac}}=-\frac{c}{12L},$$

agrees with the mass of global AdS$_3$,

$$M_{\mathrm{AdS}}=-\frac{1}{8G_3},$$

using $c=3L/(2G_3)$.

Solution

Substitute the Brown-Henneaux central charge into the cylinder vacuum energy:

$$E_{\mathrm{vac}} =-\frac{1}{12L}\frac{3L}{2G_3} =-\frac{3}{24G_3} =-\frac{1}{8G_3}.$$

This equals the mass of global AdS$_3$. The match reflects the fact that global AdS$_3$ is dual to the CFT vacuum on the cylinder, including its Casimir energy.

Exercise 3: Cardy reproduces BTZ entropy

For a rotating BTZ black hole,

$$M=\frac{r_+^2+r_-^2}{8G_3L^2}, \qquad J=\frac{r_+r_-}{4G_3L}.$$

Use

$$L_0-\frac{c}{24}=\frac{ML+J}{2}, \qquad \bar L_0-\frac{c}{24}=\frac{ML-J}{2},$$

and

$$S_{\mathrm{Cardy}} =2\pi\sqrt{\frac{c}{6}\left(L_0-\frac{c}{24}\right)} +2\pi\sqrt{\frac{c}{6}\left(\bar L_0-\frac{c}{24}\right)}$$

to derive $S_{\mathrm{Cardy}}=2\pi r_+/(4G_3)$.

Solution

First compute

$$ML+J =\frac{r_+^2+r_-^2}{8G_3L}+\frac{r_+r_-}{4G_3L} =\frac{(r_++r_-)^2}{8G_3L},$$

and

$$ML-J =\frac{(r_+-r_-)^2}{8G_3L}.$$

Therefore

$$L_0-\frac{c}{24}=\frac{(r_++r_-)^2}{16G_3L}, \qquad \bar L_0-\frac{c}{24}=\frac{(r_+-r_-)^2}{16G_3L}.$$

Using $c=3L/(2G_3)$,

$$\frac{c}{6}\left(L_0-\frac{c}{24}\right) =\frac{3L}{12G_3}\frac{(r_++r_-)^2}{16G_3L} =\frac{(r_++r_-)^2}{64G_3^2},$$

and similarly

$$\frac{c}{6}\left(\bar L_0-\frac{c}{24}\right) =\frac{(r_+-r_-)^2}{64G_3^2}.$$

Thus

$$S_{\mathrm{Cardy}} =2\pi\frac{r_++r_-}{8G_3} +2\pi\frac{r_+-r_-}{8G_3} =\frac{\pi r_+}{2G_3} =\frac{2\pi r_+}{4G_3}.$$

This is exactly the Bekenstein-Hawking entropy of the BTZ black hole.

Exercise 4: Left and right temperatures

Given

$$T_L=\frac{r_+-r_-}{2\pi L^2}, \qquad T_R=\frac{r_++r_-}{2\pi L^2},$$

show that

$$\frac{2}{T_H}=\frac{1}{T_L}+\frac{1}{T_R},$$

where

$$T_H=\frac{r_+^2-r_-^2}{2\pi L^2r_+}.$$

Solution

Compute

$$\frac{1}{T_L}+\frac{1}{T_R} =2\pi L^2\left(\frac{1}{r_+-r_-}+\frac{1}{r_++r_-}\right).$$

The expression in parentheses is

$$\frac{r_++r_-+r_+-r_-}{r_+^2-r_-^2} =\frac{2r_+}{r_+^2-r_-^2}.$$

Therefore

$$\frac{1}{T_L}+\frac{1}{T_R} =\frac{4\pi L^2r_+}{r_+^2-r_-^2} =\frac{2}{T_H}.$$

Exercise 5: The global conformal subalgebra

The Virasoro algebra is

$$[L_m,L_n] =(m-n)L_{m+n} +\frac{c}{12}m(m^2-1)\delta_{m+n,0}.$$

Show that the modes $L_{-1},L_0,L_1$ form an $sl(2,\mathbb R)$ subalgebra with no central term.

Solution

For $m=0,\pm 1$,

$$m(m^2-1)=0.$$

Therefore the central term vanishes for all brackets among $L_{-1},L_0,L_1$. The remaining commutator is

$$[L_m,L_n]=(m-n)L_{m+n}.$$

For example,

$$[L_0,L_1]=-L_1, \qquad [L_0,L_{-1}]=L_{-1}, \qquad [L_1,L_{-1}]=2L_0.$$

These are the commutation relations of $sl(2,\mathbb R)$ up to conventional choices of signs and basis. The barred modes form the second $sl(2,\mathbb R)$ factor.

Exercise 6: Chern-Simons level and central charge

In the Chern-Simons formulation of pure AdS$_3$ gravity, the level is

$$k=\frac{L}{4G_3}.$$

Show that the Brown-Henneaux central charge can be written as $c=6k$.

Solution

Substitute the level into $6k$:

$$6k=6\frac{L}{4G_3}=\frac{3L}{2G_3}.$$

This is exactly the Brown-Henneaux central charge. The relation $c=6k$ is one reason the Chern-Simons formulation is so efficient for studying AdS$_3$ boundary symmetries.

Further reading

• J. D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity.
• M. Bañados, C. Teitelboim, and J. Zanelli, The Black Hole in Three Dimensional Space Time.
• M. Bañados, M. Henneaux, C. Teitelboim, and J. Zanelli, Geometry of the 2+1 Black Hole.
• A. Strominger, Black Hole Entropy from Near-Horizon Microstates.
• J. L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories.
• S. Carlip, Conformal Field Theory, (2+1)-Dimensional Gravity, and the BTZ Black Hole.
• A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions.
• M. R. Gaberdiel and R. Gopakumar, An AdS$_3$ Dual for Minimal Model CFTs.
• J. M. Maldacena and H. Ooguri, Strings in AdS$_3$ and the $SL(2,\mathbb R)$ WZW Model. Part 1.
• O. Lunin and S. D. Mathur, Correlation Functions for $M^N/S_N$ Orbifolds.