Primaries and Descendants

The first real simplification of conformal field theory is not a formula for a correlator. It is a classification principle:

$$\boxed{\text{local operators are organized into conformal families.}}$$

Each family begins with a primary operator. Everything else in the family is obtained by differentiating it. These derivatives are called descendants. In a generic QFT, differentiating an operator gives another local operator, but there is no reason for those operators to be organized into a finite-dimensional symmetry pattern. In a CFT, they are organized by the representation theory of the conformal algebra.

For AdS/CFT, this is the first place where the CFT spectrum begins to look like a bulk spectrum. A primary operator is the boundary avatar of a single bulk species. Its descendants are not new particles; they are the same bulk excitation with different spacetime dependence, or equivalently the same conformal representation viewed at higher cylinder energy.

The local operator viewpoint

A local operator is an object that can be inserted at a point:

$$\mathcal O(x).$$

The word “operator” should not be interpreted too narrowly. Depending on context, $\mathcal O(x)$ may be a fundamental field, a composite field, a conserved current, a stress tensor, a defect operator, or a properly renormalized product of fields. In CFT we usually treat local operators as the basic observables, not as secondary objects constructed only from a Lagrangian.

The conformal algebra in Lorentzian signature is generated by

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

where $P_\mu$ generates translations, $M_{\mu\nu}$ Lorentz transformations, $D$ dilatations, and $K_\mu$ special conformal transformations. In Euclidean signature, replace the Lorentz group by $SO(d)$, and the complexified algebraic statements are the same.

The commutators most relevant for local operators are

$$[D,P_\mu]=P_\mu, \qquad [D,K_\mu]=-K_\mu, \qquad [K_\mu,P_\nu]=2\delta_{\mu\nu}D-2M_{\mu\nu},$$

with conventional factors of $i$ depending on whether one uses Hermitian or anti-Hermitian generators. These relations say that $P_\mu$ raises scaling dimension by one, while $K_\mu$ lowers scaling dimension by one.

That is the algebraic reason descendants have dimensions shifted by integers.

Scaling operators

A local operator is called a scaling operator if it transforms diagonally under dilatations. At the origin, this means

$$[D,\mathcal O(0)]=\Delta\mathcal O(0),$$

up to the same convention-dependent factors of $i$ just mentioned. The number $\Delta$ is the scaling dimension of $\mathcal O$.

Away from the origin, a scalar scaling operator transforms under a finite scale transformation as

$$\mathcal O(x)\mapsto \lambda^{-\Delta}\mathcal O(\lambda^{-1}x)$$

or, equivalently,

$$\mathcal O(\lambda x)=\lambda^{-\Delta}\mathcal O(x)$$

inside a scale-covariant correlator. A spinning operator also transforms under the rotation or Lorentz representation carried by its indices.

An operator is therefore labeled at least by

$$(\Delta,\rho,R),$$

where $\rho$ is a representation of the spacetime rotation group $SO(d)$ or Lorentz group $SO(d-1,1)$, and $R$ denotes representations of internal global symmetries. For many pages we suppress $R$ unless it matters.

Primary operators

A primary operator is a local scaling operator that is killed by the special conformal generators at the origin:

$$[K_\mu,\mathcal O(0)]=0.$$

This condition is not arbitrary. It is the conformal analogue of a highest-weight condition.

The analogy is:

$$\begin{array}{ccl} \text{highest-weight state} &:& \text{killed by raising operators},\\[2pt] \text{CFT primary} &:& \text{killed by }K_\mu,\\[2pt] \text{CFT descendants} &:& \text{generated by }P_\mu. \end{array}$$

The slight linguistic trap is that $P_\mu$ raises the dilatation eigenvalue, while $K_\mu$ lowers it. In radial quantization, $D$ is the Hamiltonian, so $P_\mu$ raises the cylinder energy and $K_\mu$ lowers it. A primary is therefore a lowest-energy state inside its conformal family.

A primary operator $\mathcal O$ is annihilated by $K_\mu$ at the origin. Acting with $P_\mu$ generates descendants such as $P_\mu\mathcal O$, $P_\mu P_\nu\mathcal O$, and so on. Under radial quantization, $D$ is the cylinder Hamiltonian, so descendants have dimensions $\Delta+n$.

Descendants

Given a primary $\mathcal O(0)$, its descendants are obtained by acting with translations:

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

In position space, translations act as derivatives, so descendants correspond to

$$\partial_{\mu_1}\cdots\partial_{\mu_n}\mathcal O(0).$$

If $\mathcal O$ has dimension $\Delta$, then a level-$n$ descendant has dimension

$$\Delta+n.$$

The descendants are not arbitrary new data. Once the primary and its transformation properties are known, the conformal algebra fixes how the descendants appear in correlators. This is why conformal blocks can sum over an entire family: a block is the contribution of one primary plus all of its descendants.

This distinction is essential. A CFT spectrum is usually listed by primary operators, not by every derivative of every operator. The complete Hilbert space contains descendants, but the independent spectral data are the primaries and their OPE coefficients.

Conformal families

The conformal family generated by a primary $\mathcal O$ is denoted

$$[\mathcal O].$$

As a vector space, it is spanned schematically by

$$[\mathcal O] = \operatorname{span}\left\{ \mathcal O, P_\mu\mathcal O, P_\mu P_\nu\mathcal O, \ldots \right\}.$$

In a unitary CFT, this family should be thought of as a representation of the conformal algebra. The primary is the lowest-weight state with respect to $D$. The descendants fill out the rest of the representation.

For a scalar primary, the first few descendants look like

$$\mathcal O, \qquad \partial_\mu\mathcal O, \qquad \partial_\mu\partial_\nu\mathcal O, \qquad \partial^2\mathcal O, \qquad \ldots.$$

For spinning primaries, the derivative indices must also be decomposed into irreducible $SO(d)$ representations. This decomposition is often tedious, but conceptually it is just the tensor product of the spin representation of $\mathcal O$ with symmetric derivative indices.

Transformation law for scalar primaries

A scalar primary has a particularly simple transformation law. Under a conformal map

$$x\mapsto x'(x),$$

with local scale factor $\Omega(x)$ defined by

$$\frac{\partial x'^\mu}{\partial x^\rho} \frac{\partial x'^\nu}{\partial x^\sigma} \delta_{\mu\nu} =\Omega(x)^2\delta_{\rho\sigma},$$

a scalar primary transforms as

$$\mathcal O'(x')=\Omega(x)^{-\Delta}\mathcal O(x).$$

This equation is one of the main reasons primaries are useful. Once the transformation law is known, conformal symmetry almost fixes two- and three-point functions.

For spinning primaries, one must also rotate the spin indices by the local orthogonal transformation associated with the conformal map. If $\mathcal O_a$ transforms in a representation $\rho$ of $SO(d)$, then schematically

$$\mathcal O'_a(x')= \Omega(x)^{-\Delta}\rho(R(x))_a{}^b\mathcal O_b(x),$$

where $R(x)$ is the local rotation part of the Jacobian.

Quasi-primaries and the special role of two dimensions

In $d>2$, the global conformal group is finite-dimensional, and the term primary usually refers to the transformation law under the full conformal group.

In two dimensions, there is a sharper distinction. The global conformal group is generated by

$$L_{-1},L_0,L_1, \qquad \bar L_{-1},\bar L_0,\bar L_1,$$

but the local conformal algebra is the full Virasoro algebra. A field can transform simply under the global subgroup but not under all Virasoro transformations. Such a field is often called quasi-primary. A true Virasoro primary obeys

$$L_n|\mathcal O\rangle=0, \qquad \bar L_n|\mathcal O\rangle=0, \qquad n>0.$$

This course will use the higher-dimensional convention unless we are explicitly in the 2D CFT modules. There, “primary” usually means Virasoro primary unless stated otherwise.

Examples

Scalar primary

The simplest example is a scalar primary $\mathcal O$ with dimension $\Delta$:

$$[D,\mathcal O(0)]=\Delta\mathcal O(0), \qquad [K_\mu,\mathcal O(0)]=0, \qquad [M_{\mu\nu},\mathcal O(0)]=0.$$

Its two-point function can be normalized as

$$\langle \mathcal O(x)\mathcal O(0)\rangle=\frac{1}{|x|^{2\Delta}}.$$

Conserved current

A conserved current $J_\mu$ in $d$ dimensions has dimension

$$\Delta_J=d-1.$$

It is a spin-one primary, except for contact-term subtleties inside Ward identities. Conservation gives

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

This is a shortening condition. The descendant $P^\mu J_\mu$ vanishes as an operator equation at separated points.

Stress tensor

The stress tensor has

$$\Delta_T=d,$$

spin two, conservation

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

and, in a CFT on flat space,

$$T^\mu{}_{\mu}=0$$

up to anomalies and contact terms. The stress tensor is not just another primary: it generates spacetime symmetries through Ward identities. Still, as an operator in a CFT spectrum, it belongs to a short conformal multiplet.

Exactly marginal operator

An exactly marginal scalar primary has

$$\Delta=d.$$

Deforming the action by

$$S\mapsto S+\lambda\int d^d x\,\mathcal O(x)$$

preserves conformal invariance if the beta function for $\lambda$ vanishes exactly. In holography, such operators are dual to massless scalar fields in AdS.

Shortening and null descendants

A generic primary generates a long conformal multiplet: none of its descendants vanish. But special values of $\Delta$ and spin can make some descendants null. When this happens, the representation shortens.

The most important examples are conserved currents. For a spin-one current,

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

removes one level-one descendant. For the stress tensor,

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

remove descendants and trace components. In two-dimensional CFT, null descendants play an even more dramatic role: they lead to differential equations for correlation functions in minimal models.

In AdS language, shortening often reflects gauge redundancy or masslessness. A conserved current is dual to a bulk gauge field. The stress tensor is dual to the graviton. BPS shortening in supersymmetric CFTs corresponds to protected supergravity multiplets.

Primaries and the OPE

The operator product expansion is organized by conformal families:

$$\mathcal O_i(x)\mathcal O_j(0) \sim \sum_k C_{ijk}\,|x|^{\Delta_k-\Delta_i-\Delta_j} \left(\mathcal O_k(0)+\text{descendants}\right).$$

The sum is over primary operators $\mathcal O_k$. The descendants are not assigned independent OPE coefficients. Their coefficients are fixed by conformal symmetry once $\Delta_k$, spin, and $C_{ijk}$ are known.

This is why the basic dynamical data of a CFT are

$$\boxed{ \text{primary spectrum }\{\Delta_i,\rho_i,R_i\} \quad\text{and}\quad \text{OPE coefficients }C_{ijk}. }$$

The descendant structure is kinematics. The primary spectrum and OPE coefficients are dynamics.

AdS/CFT checkpoint

The primary-descendant distinction becomes the boundary version of the single-particle state structure in AdS.

A scalar primary $\mathcal O$ of dimension $\Delta$ corresponds to a bulk scalar field $\phi$ with mass

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

The primary state $|\mathcal O\rangle$ is the lowest-energy state of that bulk field in global AdS. Descendants correspond to acting with spacetime momentum generators, producing higher global-AdS excitations in the same representation.

Thus one should not read

$$\partial_\mu\mathcal O$$

as a new bulk particle. It is part of the same conformal multiplet as $\mathcal O$.

For conserved currents and the stress tensor, shortening has the bulk interpretation

$$J_\mu \longleftrightarrow \text{massless gauge field}, \qquad T_{\mu\nu}\longleftrightarrow \text{graviton}.$$

This is the first clean explanation of why a CFT with global symmetry has gauge fields in the AdS dual, and why every holographic CFT has gravity in the bulk.

Common pitfalls

A derivative of a primary is usually not another primary. It is a descendant. For example, if $\mathcal O$ is primary, then $\partial_\mu\mathcal O$ is generally not primary, because $K_\nu$ does not annihilate it.

A primary is not necessarily a fundamental field in a Lagrangian. Many important primaries are composite operators, and many CFTs have no known Lagrangian at all.

A conserved current is primary only modulo contact-term subtleties in Ward identities. At separated points, it behaves as a spin-one primary of dimension $d-1$.

In two dimensions, “primary” can mean global primary or Virasoro primary. Virasoro primary is stronger.

Exercises

Exercise 1. Descendant dimensions

Let $\mathcal O$ be a primary of dimension $\Delta$. Use

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

to show that $P_\mu\mathcal O$ has dimension $\Delta+1$.

Solution

At the origin, suppose

$$D\mathcal O=\Delta\mathcal O.$$

Then

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

Using $[D,P_\mu]=P_\mu$, we get

$$D(P_\mu\mathcal O) = P_\mu(\Delta\mathcal O)+P_\mu\mathcal O =(\Delta+1)P_\mu\mathcal O.$$

Thus the level-one descendant has dimension $\Delta+1$. Repeating the argument gives dimension $\Delta+n$ for a level-$n$ descendant.

Exercise 2. Why $\partial_\mu\mathcal O$ is usually not primary

Let $\mathcal O$ be a scalar primary, so $K_\mu\mathcal O=0$ at the origin. Use

$$[K_\mu,P_\nu]=2\delta_{\mu\nu}D-2M_{\mu\nu}$$

to show that $P_\nu\mathcal O$ is generally not primary.

Solution

Compute

$$K_\mu(P_\nu\mathcal O) = P_\nu K_\mu\mathcal O+[K_\mu,P_\nu]\mathcal O.$$

The first term vanishes because $\mathcal O$ is primary. For a scalar primary, $M_{\mu\nu}\mathcal O=0$, so

$$K_\mu(P_\nu\mathcal O) =2\delta_{\mu\nu}D\mathcal O =2\Delta\delta_{\mu\nu}\mathcal O.$$

Unless $\Delta=0$, this is nonzero. Therefore $P_\nu\mathcal O$, or equivalently $\partial_\nu\mathcal O$, is generally a descendant, not a primary.

Exercise 3. Current shortening

A conserved current satisfies $\partial^\mu J_\mu=0$. Explain why this is a statement about a descendant of $J_\mu$.

Solution

The current $J_\mu$ is a spin-one operator. Acting with a translation generator gives a level-one descendant:

$$P_\nu J_\mu \longleftrightarrow \partial_\nu J_\mu.$$

Contracting the derivative index with the vector index gives

$$P^\mu J_\mu \longleftrightarrow \partial^\mu J_\mu.$$

Current conservation says that this particular descendant vanishes:

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

The multiplet is therefore shorter than a generic spin-one multiplet. This shortening is why conserved currents sit at the unitarity bound and why they are dual to massless gauge fields in AdS.

Exercise 4. Primary data versus descendant data

Why is the OPE coefficient of a descendant not independent CFT data?

Solution

The OPE is organized by conformal families. Once a primary $\mathcal O_k$ appears in the OPE

$$\mathcal O_i\times\mathcal O_j\sim C_{ijk}\mathcal O_k+ ext{descendants},$$

conformal symmetry fixes how all descendants of $\mathcal O_k$ appear. Their coefficients are determined by the primary coefficient $C_{ijk}$, the dimension $\Delta_k$, and the spin representation of $\mathcal O_k$.

Equivalently, the descendants are obtained by acting with symmetry generators. Their contribution is kinematic. The independent dynamical choice is whether the primary appears and with what coefficient.