The Conformal Algebra

The finite conformal transformations of flat space are useful, but the real engine of CFT is their Lie algebra. The algebra tells us how local operators are organized, why descendants are obtained by acting with translations, why conserved currents sit at shortening thresholds, why conformal blocks are Casimir eigenfunctions, and why AdS/CFT begins with the simple-looking identity

$$\mathrm{Conf}(\mathbb R^{1,d-1}) \simeq SO(d,2) \simeq \mathrm{Isom}(\mathrm{AdS}_{d+1}).$$

This page develops the conformal algebra in a form optimized for AdS/CFT. The main point is not just that the symmetry group is $SO(d,2)$. The main point is that CFT operator theory is representation theory of the same algebra that acts geometrically on AdS.

Conventions

For the boundary of Lorentzian $\mathrm{AdS}_{d+1}$, we take flat spacetime $\mathbb R^{1,d-1}$ with metric $\eta_{\mu\nu}$, where $\mu,\nu=0,1,\ldots,d-1$. I will write $x^2=\eta_{\mu\nu}x^\mu x^\nu$ and $b\cdot x=b_\mu x^\mu$.

The physics literature uses several sign conventions for generators. To keep the geometric part standard, start with Hermitian spacetime generators

$$P_\mu=-i\partial_\mu,$$ $$M_{\mu\nu}=i(x_\mu\partial_\nu-x_\nu\partial_\mu),$$ $$D=-ix^\rho\partial_\rho,$$ $$K_\mu=-i\left(2x_\mu x^\rho\partial_\rho-x^2\partial_\mu\right).$$

These generate translations, Lorentz transformations, dilatations, and special conformal transformations. When discussing representation theory, it is often cleaner to remove the factors of $i$ by defining

$$\mathsf P_\mu=-iP_\mu,\qquad \mathsf M_{\mu\nu}=-iM_{\mu\nu},\qquad \mathsf D=-iD,\qquad \mathsf K_\mu=-iK_\mu.$$

Then the algebra has no explicit $i$‘s. In that convention, $\mathsf P_\mu$ raises scaling dimension and $\mathsf K_\mu$ lowers it. I will use the Hermitian convention for the explicit spacetime commutators and the sans-serif convention for representation-theory statements.

For Euclidean CFT on $\mathbb R^d$, replace $SO(d,2)$ by $SO(d+1,1)$. Much of the algebraic structure is the same, but Lorentzian AdS/CFT naturally highlights $SO(d,2)$.

The conformal Killing equation

An infinitesimal coordinate transformation

$$x^\mu\mapsto x'^\mu=x^\mu+\epsilon^\mu(x)$$

is conformal if it changes the metric only by a local scale factor. To first order, this gives the conformal Killing equation

$$\partial_\mu\epsilon_\nu+ \partial_\nu\epsilon_\mu = \frac{2}{d}\eta_{\mu\nu}\partial_\rho\epsilon^\rho.$$

This equation says that the symmetric traceless part of $\partial_\mu\epsilon_\nu$ vanishes. In dimensions $d\geq 3$, its most general flat-space solution is finite-dimensional:

$$\epsilon^\mu(x) = a^\mu +\omega^\mu{}_{\nu}x^\nu +\lambda x^\mu +2(b\cdot x)x^\mu-b^\mu x^2.$$

The four terms are:

Parameter	Transformation	Number
$a^\mu$	translations	$d$
$\omega_{\mu\nu}=-\omega_{\nu\mu}$	Lorentz transformations	$d(d-1)/2$
$\lambda$	dilatations	$1$
$b^\mu$	special conformal transformations	$d$

The total number is

$$d+\frac{d(d-1)}{2}+1+d = \frac{(d+1)(d+2)}{2},$$

which is exactly the dimension of $SO(d,2)$.

This counting is already a hint. The conformal group of Lorentzian $d$-dimensional flat space is not some mysterious new object; it is the orthogonal group in a space with two extra directions.

The commutation relations

The conformal algebra is generated by

$$\{P_\mu,\ M_{\mu\nu},\ D,\ K_\mu\}.$$

The Poincare subalgebra is

$$[P_\mu,P_\nu]=0,$$ $$[M_{\mu\nu},P_\rho] =i\left(\eta_{\nu\rho}P_\mu-\eta_{\mu\rho}P_\nu\right),$$ $$[M_{\mu\nu},M_{\rho\sigma}] =i\left( \eta_{\nu\rho}M_{\mu\sigma} +\eta_{\mu\sigma}M_{\nu\rho} -\eta_{\mu\rho}M_{\nu\sigma} -\eta_{\nu\sigma}M_{\mu\rho} \right).$$

The dilatation generator commutes with Lorentz transformations and assigns weights to $P_\mu$ and $K_\mu$:

$$[D,M_{\mu\nu}]=0,$$ $$[D,P_\mu]=iP_\mu, \qquad [D,K_\mu]=-iK_\mu.$$

Special conformal transformations commute among themselves,

$$[K_\mu,K_\nu]=0,$$

and transform as Lorentz vectors,

$$[M_{\mu\nu},K_\rho] =i\left(\eta_{\nu\rho}K_\mu-\eta_{\mu\rho}K_\nu\right).$$

The most important mixed commutator is

$$[K_\mu,P_\nu] =2i\left(\eta_{\mu\nu}D-M_{\mu\nu}\right).$$

This last relation is where most of the representation theory begins. A primary operator is annihilated by $K_\mu$ at the origin; descendants are obtained by acting with $P_\mu$; and the mixed commutator controls the norms of descendants. Those norms lead directly to unitarity bounds.

The three-grading by dilatations

In the representation-theory convention with generators $\mathsf G=-iG$, the relevant commutators become

$$[\mathsf D,\mathsf P_\mu]=\mathsf P_\mu, \qquad [\mathsf D,\mathsf K_\mu]=-\mathsf K_\mu,$$ $$[\mathsf K_\mu,\mathsf P_\nu] =2\left(\eta_{\mu\nu}\mathsf D-\mathsf M_{\mu\nu}\right).$$

Thus the algebra has a natural three-grading:

$$\mathfrak{so}(d,2) = \mathfrak g_{-1}\oplus \mathfrak g_0\oplus \mathfrak g_{+1},$$

where

$$\mathfrak g_{-1}=\mathrm{span}\{\mathsf K_\mu\}, \qquad \mathfrak g_0=\mathrm{span}\{\mathsf D,\mathsf M_{\mu\nu}\}, \qquad \mathfrak g_{+1}=\mathrm{span}\{\mathsf P_\mu\}.$$

The conformal algebra is naturally organized by dilatation weight. In radial quantization, $P_\mu$ generates descendants and $K_\mu$ annihilates primaries at the origin. The same algebra is $\mathfrak{so}(d,2)$, the isometry algebra of $\mathrm{AdS}_{d+1}$.

The grading means

$$[\mathfrak g_m,\mathfrak g_n]\subseteq \mathfrak g_{m+n},$$

with $\mathfrak g_{\pm 2}=0$. In particular,

$$[\mathsf P,\mathsf P]=0, \qquad [\mathsf K,\mathsf K]=0, \qquad [\mathsf K,\mathsf P]\subseteq \mathfrak g_0.$$

This is the algebraic reason CFT representations look like highest-weight or lowest-weight representations. At the origin, $\mathsf K_\mu$ is a lowering operator, so a unitary representation cannot keep lowering forever. It must start from a primary.

Primaries and descendants

A local operator at the origin is classified by its transformation under $\mathfrak g_0$, namely by its scaling dimension and Lorentz representation. A primary operator $\mathcal O_a(0)$ obeys

$$[\mathsf K_\mu,\mathcal O_a(0)]=0,$$ $$[\mathsf D,\mathcal O_a(0)]=\Delta\mathcal O_a(0),$$ $$[\mathsf M_{\mu\nu},\mathcal O_a(0)] =(\Sigma_{\mu\nu})_a{}^b\mathcal O_b(0),$$

where $a,b$ are spin indices and $\Sigma_{\mu\nu}$ are the Lorentz generators in the representation carried by the operator.

Descendants are obtained by acting with translations:

$$\mathsf P_{\mu_1}\cdots \mathsf P_{\mu_n}\mathcal O_a(0).$$

Because

$$[\mathsf D,\mathsf P_\mu]=\mathsf P_\mu,$$

a level-$n$ descendant has dimension $\Delta+n$. This is why a conformal multiplet is discrete above its primary even when the CFT is interacting.

Away from the origin, the transformation of a scalar primary is fixed by translating the origin result. In infinitesimal form,

$$\delta\mathcal O(x) =-\epsilon^\mu(x)\partial_\mu\mathcal O(x) -\frac{\Delta}{d}\left(\partial_\mu\epsilon^\mu(x)\right)\mathcal O(x)$$

for a scalar primary. For spinning primaries, one adds a local Lorentz rotation term proportional to $\partial_{[\mu}\epsilon_{\nu]}\Sigma^{\mu\nu}$.

This formula is worth digesting. It says that a primary is not just any local operator. It transforms covariantly under the local scale factor generated by a conformal transformation. Descendants transform with extra inhomogeneous terms because derivatives of primaries are not usually primaries.

Repackaging the algebra as $\mathfrak{so}(d,2)$

Introduce auxiliary indices

$$A,B=-1,0,1,\ldots,d$$

and generators $J_{AB}=-J_{BA}$ of $\mathfrak{so}(d,2)$ satisfying

$$[J_{AB},J_{CD}] =i\left( \eta_{BC}J_{AD} +\eta_{AD}J_{BC} -\eta_{AC}J_{BD} -\eta_{BD}J_{AC} \right),$$

where the ambient metric has two timelike directions. The conformal generators can be embedded as

$$J_{\mu\nu}=M_{\mu\nu},$$ $$J_{-1,0}=D,$$ $$J_{-1,\mu}=\frac{1}{2}(P_\mu-K_\mu), \qquad J_{0,\mu}=\frac{1}{2}(P_\mu+K_\mu).$$

This change of basis turns the conformal commutators into the standard orthogonal Lie algebra. Conceptually, it says that translations and special conformal transformations are not unrelated miracles. They are rotations involving the two extra embedding directions.

This is also the first place where the future AdS story becomes unavoidable. The conformal algebra is already the algebra of rotations in $\mathbb R^{d,2}$, and $\mathrm{AdS}_{d+1}$ is naturally defined as a hyperboloid inside that same space.

The AdS isometry algebra

Embed $\mathrm{AdS}_{d+1}$ into $\mathbb R^{d,2}$ with coordinates $X^A$ and metric of signature $(d,2)$ by

$$-X_{-1}^2-X_0^2+X_1^2+\cdots+X_d^2=-R^2.$$

The transformations preserving the ambient quadratic form are $SO(d,2)$. Since they also preserve the hyperboloid, they are isometries of $\mathrm{AdS}_{d+1}$:

$$\mathrm{Isom}(\mathrm{AdS}_{d+1})=SO(d,2).$$

This gives the group-theoretic core of AdS/CFT:

$$\boxed{\text{boundary conformal symmetry} = \text{bulk AdS isometry}.}$$

In Poincare coordinates,

$$ds^2=R^2\frac{dz^2+\eta_{\mu\nu}dx^\mu dx^\nu}{z^2}, \qquad z>0,$$

the boundary is at $z\to 0$. The conformal generators act on the boundary coordinates $x^\mu$ in the expected way. In the bulk, they extend to AdS Killing vectors. For example, dilatations act as

$$x^\mu\mapsto \lambda x^\mu, \qquad z\mapsto \lambda z.$$

The radial coordinate transforms like an energy scale. This is the seed of the relation between radial motion in AdS and RG scale in the CFT.

Special conformal transformations also extend into the bulk. Infinitesimally,

$$\delta_b x^\mu =2(b\cdot x)x^\mu-b^\mu(x^2+z^2),$$ $$\delta_b z=2(b\cdot x)z.$$

At the boundary $z=0$, this reduces to the ordinary boundary special conformal transformation

$$\delta_b x^\mu =2(b\cdot x)x^\mu-b^\mu x^2.$$

The extra $-b^\mu z^2$ term is precisely what is needed to preserve the AdS metric in the interior.

The quadratic Casimir

The quadratic Casimir of the conformal algebra is

$$\mathcal C_2=\frac{1}{2}J_{AB}J^{AB}.$$

For a primary operator of scaling dimension $\Delta$ in a symmetric traceless spin-$\ell$ representation, the eigenvalue is

$$C_2(\Delta,\ell) =\Delta(\Delta-d)+\ell(\ell+d-2).$$

This formula appears everywhere:

• conformal blocks are eigenfunctions of the conformal Casimir;
• bulk fields in AdS furnish representations of the same algebra;
• the scalar mass-dimension relation is the spin-zero version.

For a scalar primary dual to a scalar bulk field,

$$m^2R^2=\Delta(\Delta-d).$$

Solving for $\Delta$ gives

$$\Delta_\pm=\frac{d}{2}\pm\sqrt{\frac{d^2}{4}+m^2R^2}.$$

Already at the algebraic level, bulk mass is not merely a parameter in a wave equation. It is the label of an $SO(d,2)$ representation, hence the label of a CFT conformal family.

Conserved currents and shortening

Some conformal multiplets are shorter than a generic primary multiplet. The simplest examples are conserved currents.

A spin-one primary current obeys

$$\partial^\mu J_\mu=0.$$

In representation theory, this means that a descendant is null. The current multiplet is shorter than a generic spin-one multiplet. In a unitary CFT, this shortening occurs at the protected dimension

$$\Delta_J=d-1.$$

Similarly, the stress tensor obeys

$$\partial^\mu T_{\mu\nu}=0, \qquad T^\mu{}_{\mu}=0,$$

and has protected dimension

$$\Delta_T=d.$$

In AdS/CFT, these protected CFT operators identify massless bulk gauge fields:

$$J_\mu \longleftrightarrow A_M, \qquad T_{\mu\nu}\longleftrightarrow g_{MN}.$$

The logic is symmetry-based. Conserved currents generate global symmetries in the CFT; in AdS, the corresponding fields are gauge fields. The stress tensor generates spacetime symmetry in the CFT; in AdS, the corresponding field is the graviton.

The special case $d=2$

For $d\geq 3$, the flat-space conformal algebra is finite-dimensional. In two dimensions, the local conformal algebra becomes infinite-dimensional, and after quantization one obtains two Virasoro algebras. But the global part is still finite:

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

in Lorentzian signature. In Euclidean signature, the global group is related to $SL(2,\mathbb C)$ acting by Mobius transformations on the Riemann sphere.

This distinction is crucial. The global conformal algebra is the symmetry matching the isometries of $\mathrm{AdS}_3$. The Virasoro algebra is a much stronger asymptotic symmetry structure, and its central charge is what eventually enters the Brown-Henneaux and Cardy stories.

What this page buys us

The conformal algebra gives a compact answer to several questions that otherwise look unrelated.

First, why do local operators come in families? Because $\mathsf P_\mu$ generates descendants from a primary annihilated by $\mathsf K_\mu$.

Second, why are scaling dimensions like energies? Because after radial quantization, $\mathsf D$ becomes the Hamiltonian on the cylinder.

Third, why does AdS have one extra radial direction? Because the CFT dilatation generator acts geometrically in the bulk as a rescaling of both boundary coordinates and the AdS radial coordinate.

Fourth, why does the scalar dictionary contain $m^2R^2=\Delta(\Delta-d)$? Because both sides are labels of the same $SO(d,2)$ representation.

The next pages will turn this algebra into geometry, first through conformal compactification and the cylinder, and then through embedding-space methods.

Exercises

Exercise 1: Count the conformal generators

Use the conformal Killing equation in $d\geq 3$ to argue that the conformal algebra has

$$\frac{(d+1)(d+2)}{2}$$

generators.

Solution

For $d\geq 3$, the general infinitesimal conformal Killing vector is

$$\epsilon^\mu(x) = a^\mu +\omega^\mu{}_{\nu}x^\nu +\lambda x^\mu +2(b\cdot x)x^\mu-b^\mu x^2.$$

The parameters are:

$$a^\mu: d, \qquad \omega_{\mu\nu}=-\omega_{\nu\mu}: \frac{d(d-1)}{2}, \qquad \lambda: 1, \qquad b^\mu:d.$$

Therefore the total number is

$$d+\frac{d(d-1)}{2}+1+d =\frac{d^2+3d+2}{2} =\frac{(d+1)(d+2)}{2}.$$

This equals $\dim SO(d,2)$, since $SO(N)$ has dimension $N(N-1)/2$ and here $N=d+2$.

Exercise 2: Show that translations raise dimension

In the representation-theory convention, suppose

$$[\mathsf D,\mathsf P_\mu]=\mathsf P_\mu, \qquad [\mathsf D,\mathcal O(0)]=\Delta\mathcal O(0).$$

Show that the descendant $[\mathsf P_\mu,\mathcal O(0)]$ has dimension $\Delta+1$.

Solution

Use the Jacobi identity:

$$[\mathsf D,[\mathsf P_\mu,\mathcal O]] = [[\mathsf D,\mathsf P_\mu],\mathcal O] +[\mathsf P_\mu,[\mathsf D,\mathcal O]].$$

Substituting the commutators gives

$$[\mathsf D,[\mathsf P_\mu,\mathcal O]] = [\mathsf P_\mu,\mathcal O] +\Delta[\mathsf P_\mu,\mathcal O] =(\Delta+1)[\mathsf P_\mu,\mathcal O].$$

Thus $\mathsf P_\mu\mathcal O$ is a level-one descendant of dimension $\Delta+1$.

Exercise 3: Recover the scalar AdS mass-dimension relation

A scalar conformal family has quadratic Casimir eigenvalue

$$C_2(\Delta,0)=\Delta(\Delta-d).$$

A scalar field in $\mathrm{AdS}_{d+1}$ has the same $SO(d,2)$ Casimir eigenvalue $m^2R^2$. Find $\Delta$.

Solution

Equating the two eigenvalues gives

$$m^2R^2=\Delta(\Delta-d).$$

This is a quadratic equation:

$$\Delta^2-d\Delta-m^2R^2=0.$$

Solving gives

$$\Delta_\pm =\frac{d}{2}\pm\sqrt{\frac{d^2}{4}+m^2R^2}.$$

The standard quantization usually chooses $\Delta_+$, while in an allowed mass window above the Breitenlohner-Freedman bound there can also be an alternate quantization associated with $\Delta_-$.

Exercise 4: Check the boundary limit of a bulk special conformal transformation

In Poincare coordinates, the AdS special conformal Killing vector has infinitesimal action

$$\delta_b x^\mu=2(b\cdot x)x^\mu-b^\mu(x^2+z^2),$$ $$\delta_b z=2(b\cdot x)z.$$

Show that it reduces to the ordinary boundary special conformal transformation as $z\to0$.

Solution

At the boundary, set $z=0$ in the transformation of $x^\mu$:

$$\delta_b x^\mu =2(b\cdot x)x^\mu-b^\mu x^2.$$

This is exactly the infinitesimal special conformal transformation on $\mathbb R^{1,d-1}$.

The radial variation is

$$\delta_b z=2(b\cdot x)z,$$

so if $z=0$, then $\delta_b z=0$. Thus the boundary is mapped to itself, as required for a bulk isometry inducing a boundary conformal transformation.

Exercise 5: Why is the stress tensor dimension protected?

Use the fact that $T_{\mu\nu}$ is conserved and that the conserved translation charge is

$$P_\nu=\int d^{d-1}x\,T_{0\nu}$$

to argue dimensionally that $T_{\mu\nu}$ has scaling dimension $d$.

Solution

The translation generator $P_\nu$ has scaling dimension $1$, because it has the same dimension as a derivative:

$$[P_\nu]=1.$$

The spatial integral has dimension $-(d-1)$, so if $T_{0\nu}$ has dimension $\Delta_T$, then

$$[P_\nu]=\Delta_T-(d-1).$$

Equating this to $1$ gives

$$\Delta_T-(d-1)=1,$$

hence

$$\Delta_T=d.$$

This is consistent with conservation $\partial^\mu T_{\mu\nu}=0$, which is a shortening condition of the conformal multiplet.

Further reading

For the classic two-dimensional perspective, see Di Francesco, Mathieu, and Senechal, Chapter 4 for global conformal invariance and Chapter 5 for the special enhancement in two dimensions. For a modern higher-dimensional CFT treatment, see Rychkov’s lectures and Simmons-Duffin’s TASI lectures. For the AdS/CFT use of $SO(d,2)$, the essential next step is to study AdS as a hyperboloid embedded in $\mathbb R^{d,2}$.