Casimir and Conformal Families

A primary operator is not a lonely object. Once a primary exists, conformal symmetry forces an infinite tower of descendants to exist with it. This tower is called a conformal family or conformal multiplet. The family is the local-operator version of one representation of the conformal algebra.

The organizing invariant of this representation is the quadratic conformal Casimir. It is the same number on the primary and on every descendant. This is why conformal blocks are not arbitrary functions: they are eigenfunctions of a second-order differential operator whose eigenvalue is fixed by the exchanged primary.

The basic slogan is

$$\boxed{ \text{one primary }\mathcal O_{\Delta,\rho} \quad\Longrightarrow\quad \text{one family }[\mathcal O] \quad\Longrightarrow\quad \text{one Casimir eigenvalue} }$$

where $\Delta$ is the scaling dimension and $\rho$ is the $SO(d)$ spin representation.

For AdS/CFT this is not just representation-theory bookkeeping. The same group $SO(d,2)$ acts as the conformal group of the boundary CFT and as the isometry group of $AdS_{d+1}$. The CFT Casimir is therefore the boundary form of the bulk wave-equation label. For a scalar field in $AdS_{d+1}$,

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

which is exactly the scalar part of the conformal Casimir eigenvalue.

Conformal families

Let $\mathcal O_{\Delta,\rho}(x)$ be a primary operator. In radial quantization it creates the state

$$|\mathcal O\rangle=\mathcal O(0)|0\rangle.$$

The primary conditions are

$$D|\mathcal O\rangle=\Delta|\mathcal O\rangle, \qquad K_\mu|\mathcal O\rangle=0.$$

Descendants are obtained by acting with momenta:

$$P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle, \qquad n=0,1,2,\ldots.$$

The conformal family is the span of all such states:

$$[\mathcal O] = \operatorname{span}\left\{ P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle \right\}.$$

Since

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

a level-$n$ descendant has dimension

$$\Delta+n.$$

In position space, acting with $P_\mu$ corresponds to taking a derivative:

$$P_\mu \quad\longleftrightarrow\quad \partial_\mu.$$

Thus the local operators in the conformal family are schematically

$$\mathcal O, \qquad \partial_\mu\mathcal O, \qquad \partial_{\mu_1}\partial_{\mu_2}\mathcal O, \qquad \ldots.$$

When $\mathcal O$ carries spin, the descendants are reorganized into irreducible $SO(d)$ representations. For example, a level-one descendant of a symmetric traceless spin-$\ell$ primary is obtained from

$$[\ell]\otimes[1],$$

which decomposes into spin $\ell+1$, spin $\ell-1$, and mixed-symmetry pieces. The divergence descendant is the spin $\ell-1$ part.

A conformal family $[\mathcal O_{\Delta,\rho}]$ is generated from one primary by repeated action of $P_\mu$. The quadratic Casimir $\mathcal C_2$ has the same eigenvalue on every descendant. In a four-point function, the conformal block $G_{\Delta,\ell}$ is the contribution of the whole family and solves a Casimir eigenvalue equation.

Families, multiplets, and modules

The words family, multiplet, and representation are often used almost interchangeably. The distinction is mostly one of viewpoint.

language	meaning
conformal representation	the abstract representation of the conformal algebra
conformal multiplet	the states in radial quantization
conformal family	the corresponding local operators
primary	lowest-weight state, annihilated by $K_\mu$
descendant	state obtained by acting with $P_\mu$

A generic family is called a long multiplet. At special values of $\Delta$, some descendant can become null. Then the physical irreducible representation is obtained by quotienting out the null submodule. This gives a short multiplet.

The previous page gave the most important examples:

$$\Delta=\frac{d-2}{2} \quad\Longrightarrow\quad \partial^2\mathcal O=0 \qquad \text{for a scalar,}$$

and

$$\Delta=\ell+d-2 \quad\Longrightarrow\quad \partial^{\mu_1}\mathcal O_{\mu_1\cdots\mu_\ell}=0 \qquad \text{for a symmetric traceless spin-$\ell$ operator.}$$

The first is a free equation of motion. The second is a conservation equation. Both are statements about null descendants inside a conformal family.

The quadratic conformal Casimir

The conformal algebra is generated by

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

We use radial-quantization conventions with

$$[D,P_\mu]=P_\mu, \qquad [D,K_\mu]=-K_\mu,$$

and

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

The quadratic Casimir may be written as

$$\boxed{ \mathcal C_2 = D^2 - \frac12\left(P_\mu K_\mu+K_\mu P_\mu\right) + \frac12M_{\mu\nu}M_{\mu\nu}. }$$

It commutes with every conformal generator:

$$[\mathcal C_2,D]=0, \qquad [\mathcal C_2,P_\mu]=0, \qquad [\mathcal C_2,K_\mu]=0, \qquad [\mathcal C_2,M_{\mu\nu}]=0.$$

Equivalently, using

$$K_\mu P_\mu=P_\mu K_\mu+2dD,$$

we can write

$$\mathcal C_2 = D(D-d)+\frac12M_{\mu\nu}M_{\mu\nu}-P_\mu K_\mu.$$

This second form makes the eigenvalue on a primary transparent. Since

$$K_\mu|\mathcal O\rangle=0,$$

we get

$$P_\mu K_\mu|\mathcal O\rangle=0.$$

Therefore

$$\mathcal C_2|\mathcal O\rangle = \left[D(D-d)+\frac12M_{\mu\nu}M_{\mu\nu}\right]|\mathcal O\rangle.$$

The dilatation part gives

$$D(D-d)|\mathcal O\rangle = \Delta(\Delta-d)|\mathcal O\rangle.$$

The rotation part gives the ordinary $SO(d)$ quadratic Casimir of the spin representation $\rho$:

$$\frac12M_{\mu\nu}M_{\mu\nu}|\mathcal O\rangle =C_2^{SO(d)}(\rho)|\mathcal O\rangle.$$

Thus the conformal Casimir eigenvalue is

$$\boxed{ C_{\Delta,\rho} = \Delta(\Delta-d)+C_2^{SO(d)}(\rho). }$$

For a symmetric traceless spin-$\ell$ representation,

$$C_2^{SO(d)}(\ell)=\ell(\ell+d-2),$$

so

$$\boxed{ C_{\Delta,\ell} = \Delta(\Delta-d)+\ell(\ell+d-2). }$$

This is one of the most frequently used formulas in modern CFT.

Why every descendant has the same Casimir

Because $\mathcal C_2$ commutes with $P_\mu$, it has the same eigenvalue on descendants. If

$$\mathcal C_2|\mathcal O\rangle =C_{\Delta,\rho}|\mathcal O\rangle,$$

then

$$\begin{aligned} \mathcal C_2 P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle &=P_{\mu_1}\cdots P_{\mu_n}\mathcal C_2|\mathcal O\rangle \\ &=C_{\Delta,\rho}P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle. \end{aligned}$$

The descendants have different scaling dimensions and different $SO(d)$ tensor decompositions, but they share the same conformal Casimir eigenvalue. This is exactly what it means for them to belong to one conformal representation.

Useful examples

Scalar primary

For a scalar primary, $\rho$ is the trivial representation, so

$$C_2^{SO(d)}(0)=0.$$

Therefore

$$C_{\Delta,0}=\Delta(\Delta-d).$$

This is the same expression that appears in the scalar AdS mass-dimension relation,

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

The equality is not accidental. The scalar Laplacian on AdS is the geometric realization of the $SO(d,2)$ Casimir.

Conserved current

For a conserved current,

$$\Delta_J=d-1, \qquad \ell_J=1.$$

Thus

$$C_J=(d-1)(-1)+1(d-1)=0.$$

This is a useful warning: the quadratic Casimir is powerful, but it is not a complete operator label. The identity representation also has $C_2=0$, but a conserved current and the identity are obviously different representations.

Stress tensor

For the stress tensor,

$$\Delta_T=d, \qquad \ell_T=2.$$

Therefore

$$C_T=d(d-d)+2(2+d-2)=2d.$$

The stress-tensor family is universal in any local CFT. In AdS/CFT it is the boundary family associated with the graviton.

Large-$N$ double-trace operators

In a large-$N$ CFT, suppose $\mathcal O$ is a scalar single-trace primary with dimension $\Delta_{\mathcal O}$. Double-trace primaries have schematic form

$$[\mathcal O\mathcal O]_{n,\ell} \sim \mathcal O\,\partial^{2n}\partial_{(\mu_1}\cdots\partial_{\mu_\ell)}\mathcal O - \text{traces},$$

with leading dimensions

$$\Delta_{n,\ell}^{(0)}=2\Delta_{\mathcal O}+2n+\ell.$$

Their leading Casimir eigenvalues are

$$C_{n,\ell}^{(0)} = \Delta_{n,\ell}^{(0)}\left(\Delta_{n,\ell}^{(0)}-d\right) + \ell(\ell+d-2).$$

These are the CFT counterparts of two-particle states in global AdS.

Shadow symmetry of the Casimir

The quadratic Casimir satisfies

$$C_{\Delta,\ell}=C_{d-\Delta,\ell}.$$

Indeed,

$$(d-\Delta)((d-\Delta)-d) =(d-\Delta)(-\Delta) =\Delta(\Delta-d).$$

The representation with dimension

$$\widetilde\Delta=d-\Delta$$

is called the shadow representation. The Casimir equation alone cannot distinguish a block from its shadow solution. The OPE boundary condition does.

In the OPE limit $x_{12}\to0$, the cross-ratio

$$u= \frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}$$

goes to zero. A physical block for an exchanged operator of dimension $\Delta$ behaves as

$$G_{\Delta,\ell}(u,v) \sim u^{\Delta/2} \times \text{spin-$\ell$ angular structure}.$$

The shadow solution behaves instead as

$$G_{d-\Delta,\ell}(u,v) \sim u^{(d-\Delta)/2} \times \text{spin-$\ell$ angular structure}.$$

Thus

$$\boxed{ \text{Casimir equation} + \text{OPE boundary condition} \quad\Longrightarrow\quad \text{physical conformal block}. }$$

The OPE sums over families

The operator product expansion does not merely sum over primary operators. It sums over conformal families. For two scalar primaries,

$$\phi_i(x)\phi_j(0) \sim \sum_{\mathcal O} \lambda_{ij\mathcal O} |x|^{\Delta_{\mathcal O}-\Delta_i-\Delta_j} \left[ \mathcal O(0)+\text{descendants} \right].$$

The sum is over primary operators $\mathcal O$. Once the primary and coefficient $\lambda_{ij\mathcal O}$ are specified, conformal symmetry fixes the relative coefficients of all descendants.

This is why the data of a CFT can be listed as

$$\left\{\Delta_i,\rho_i,\lambda_{ijk}\right\},$$

rather than as an infinite list of unrelated derivative operators.

Conformal blocks as family contributions

Consider a four-point function of scalar primaries. Conformal symmetry lets us write it as a kinematic prefactor times a function of cross-ratios:

$$\langle \phi_1(x_1)\phi_2(x_2)\phi_3(x_3)\phi_4(x_4) \rangle = \text{prefactor}\times\mathcal G(u,v),$$

where

$$u= \frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}, \qquad v= \frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}.$$

In the $12\to34$ OPE channel,

$$\mathcal G(u,v) = \sum_{\mathcal O} \lambda_{12\mathcal O}\lambda_{34\mathcal O} G_{\Delta,\ell}^{12,34}(u,v).$$

The function $G_{\Delta,\ell}^{12,34}(u,v)$ is the conformal block for the family $[\mathcal O_{\Delta,\ell}]$.

A conformal block is therefore

$$G_{\Delta,\ell} = \text{primary contribution} + \text{level 1 descendants} + \text{level 2 descendants} + \cdots.$$

It is the complete contribution of one conformal family.

The Casimir equation for blocks

Let $J_i^{AB}$ be the conformal generators acting on the point $x_i$. The total generator acting on the pair $(1,2)$ is

$$J_{12}^{AB}=J_1^{AB}+J_2^{AB}.$$

The corresponding quadratic Casimir is

$$\mathcal C_{12} = \frac12J_{12}^{AB}J_{12,AB}.$$

When the $12$ OPE exchanges the family $[\mathcal O_{\Delta,\ell}]$, the pair $(1,2)$ transforms in the representation of $\mathcal O$. Therefore the block satisfies

$$\boxed{ \mathcal C_{12}G_{\Delta,\ell}^{12,34}(u,v) = C_{\Delta,\ell}G_{\Delta,\ell}^{12,34}(u,v). }$$

This equation is the origin of the Casimir method for conformal blocks. In practice $\mathcal C_{12}$ becomes a second-order differential operator in the cross-ratios $u,v$, or equivalently in variables $z,\bar z$ defined by

$$u=z\bar z, \qquad v=(1-z)(1-\bar z).$$

The explicit differential operator is not needed yet. The structural point is enough:

$$\boxed{ \text{conformal blocks are Casimir eigenfunctions labeled by }(\Delta,\ell). }$$

This is directly analogous to ordinary angular momentum. Spherical harmonics are eigenfunctions of the $SO(3)$ Casimir $L^2$; conformal blocks are eigenfunctions of the $SO(d,2)$ Casimir.

Normalization and OPE behavior

The Casimir equation fixes the block only after choosing a normalization and imposing the physical OPE behavior. A common normalization is

$$G_{\Delta,\ell}(u,v) \sim u^{\Delta/2} \left[ \text{spin-$\ell$ angular structure} +O(u) \right] \qquad (u\to0).$$

For identical external scalars, the leading angular structure is often written using a Gegenbauer polynomial,

$$G_{\Delta,\ell}(u,v) \sim u^{\Delta/2}C_\ell^{(d/2-1)}(\cos\theta),$$

where $\theta$ is the angle between the two OPE pairs in the small-$u$ limit.

The conceptual content is convention-independent:

$$\text{the leading power }u^{\Delta/2}\text{ knows the dimension,} \qquad \text{the angular dependence knows the spin.}$$

Two-dimensional comment

In two dimensions, global conformal symmetry factorizes into holomorphic and antiholomorphic parts. A primary is labeled by

$$(h,\bar h), \qquad \Delta=h+\bar h, \qquad s=h-\bar h.$$

The global quadratic Casimirs are essentially

$$h(h-1), \qquad \bar h(\bar h-1).$$

But two-dimensional CFT usually has the full Virasoro algebra. A global conformal family is generated by

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

whereas a Virasoro family is generated by

$$L_{-n},\quad \bar L_{-n}, \qquad n\geq1.$$

Therefore

$$\boxed{ \text{global family} \subset \text{Virasoro family}. }$$

This is why Virasoro blocks in $d=2$ are much richer than global conformal blocks. In AdS$_3$/CFT$_2$, this distinction becomes the statement that boundary gravitons are encoded in Virasoro descendants.

AdS/CFT checkpoint

The representation-theoretic picture of this page is almost the AdS/CFT dictionary in miniature.

In global $AdS_{d+1}$, the isometry group is $SO(d,2)$, the same as the boundary conformal group. A one-particle bulk state forms a lowest-weight representation:

$$\text{bulk one-particle representation of }SO(d,2) \quad\longleftrightarrow\quad \text{CFT conformal family }[\mathcal O].$$

The CFT primary corresponds to the lowest-energy bulk state:

$$E_0=\Delta.$$

The CFT descendants correspond to global AdS excitations:

$$P_\mu|\mathcal O\rangle \quad\longleftrightarrow\quad \text{one unit of global AdS excitation}.$$

For a scalar field,

$$C_{\Delta,0}=\Delta(\Delta-d)=m^2R^2.$$

For spinning fields, the Casimir includes the spin contribution and encodes the corresponding $SO(d,2)$ representation. At the level of four-point functions, the Casimir equation for a conformal block is the boundary counterpart of a bulk wave equation for an exchanged field.

This is why conformal blocks are central in holography. They are not merely convenient basis functions. They are the boundary representation-theoretic imprint of exchanging a definite AdS representation.

Common pitfalls

Pitfall 1: thinking the primary alone is exchanged

The OPE and the conformal block decomposition exchange an entire family:

$$\mathcal O+\partial\mathcal O+\partial^2\mathcal O+\cdots.$$

The primary labels the family, but the descendants are physically present and required by symmetry.

Pitfall 2: using $C_{\Delta,\ell}$ as a complete label

The quadratic Casimir is not always unique. Shadow-related representations share it, and different physical operators can have the same $(\Delta,\ell)$ or the same Casimir. A full CFT label includes spin representation, global-symmetry representation, degeneracy label, and OPE coefficients.

Pitfall 3: forgetting shortening

A conserved current is not a generic spin-one family. A stress tensor is not a generic spin-two family. Conservation removes null descendants. This affects state counting, conformal blocks, and holographic interpretation.

Pitfall 4: confusing global and Virasoro families in $d=2$

A 2D global family is generated by $L_{-1}$ and $\bar L_{-1}$. A Virasoro family is generated by all $L_{-n}$ and $\bar L_{-n}$ with $n\geq1$. The Virasoro family is much larger.

Exercises

Exercise 1: Casimir eigenvalue of a scalar primary

Use

$$\mathcal C_2 = D(D-d)+\frac12M_{\mu\nu}M_{\mu\nu}-P_\mu K_\mu$$

to compute the quadratic Casimir eigenvalue of a scalar primary of dimension $\Delta$.

Solution

For a scalar primary,

$$K_\mu|\mathcal O\rangle=0, \qquad D|\mathcal O\rangle=\Delta|\mathcal O\rangle, \qquad M_{\mu\nu}|\mathcal O\rangle=0.$$

Therefore

$$P_\mu K_\mu|\mathcal O\rangle=0,$$

and

$$\mathcal C_2|\mathcal O\rangle =D(D-d)|\mathcal O\rangle =\Delta(\Delta-d)|\mathcal O\rangle.$$

Thus

$$C_{\Delta,0}=\Delta(\Delta-d).$$

Exercise 2: descendants have the same Casimir

Suppose

$$\mathcal C_2|\mathcal O\rangle=C_{\Delta,\rho}|\mathcal O\rangle.$$

Show that $P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle$ has the same eigenvalue.

Solution

The quadratic Casimir commutes with every conformal generator, in particular

$$[\mathcal C_2,P_\mu]=0.$$

Therefore

$$\begin{aligned} \mathcal C_2P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle &=P_{\mu_1}\cdots P_{\mu_n}\mathcal C_2|\mathcal O\rangle \\ &=C_{\Delta,\rho}P_{\mu_1}\cdots P_{\mu_n}|\mathcal O\rangle. \end{aligned}$$

All descendants in one conformal family have the same quadratic Casimir eigenvalue.

Exercise 3: Casimir values for special operators

For a symmetric traceless spin-$\ell$ primary,

$$C_{\Delta,\ell}=\Delta(\Delta-d)+\ell(\ell+d-2).$$

Compute the value for the identity operator, a conserved current $J_\mu$, and the stress tensor $T_{\mu\nu}$.

Solution

The identity has

$$\Delta=0, \qquad \ell=0,$$

so

$$C_{\mathbf 1}=0.$$

A conserved current has

$$\Delta_J=d-1, \qquad \ell_J=1.$$

Hence

$$C_J=(d-1)(-1)+(d-1)=0.$$

The stress tensor has

$$\Delta_T=d, \qquad \ell_T=2.$$

Therefore

$$C_T=d(d-d)+2(2+d-2)=2d.$$

So

$$C_{\mathbf 1}=0, \qquad C_J=0, \qquad C_T=2d.$$

Exercise 4: shadow degeneracy

Show that

$$C_{\Delta,\ell}=C_{d-\Delta,\ell}.$$

Why does this not mean that $\Delta$ and $d-\Delta$ describe the same physical operator in a given CFT?

Solution

The spin term is unchanged. The dimension-dependent term obeys

$$(d-\Delta)((d-\Delta)-d) =(d-\Delta)(-\Delta) =\Delta(\Delta-d).$$

Therefore

$$C_{d-\Delta,\ell}=C_{\Delta,\ell}.$$

The two representations solve the same Casimir equation, but their OPE behavior differs:

$$G_{\Delta,\ell}\sim u^{\Delta/2}, \qquad G_{d-\Delta,\ell}\sim u^{(d-\Delta)/2}.$$

The physical OPE expansion selects the behavior associated with the actual operator dimension in the spectrum. The other solution is the shadow solution.

Exercise 5: scalar mass-dimension relation in AdS

Assume a scalar bulk field in $AdS_{d+1}$ is dual to a scalar primary of dimension $\Delta$. Starting from

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

solve for $\Delta$.

Solution

The equation is quadratic:

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

Solving gives

$$\Delta_\pm = \frac d2 \pm \sqrt{\frac{d^2}{4}+m^2R^2}.$$

The two roots obey

$$\Delta_++\Delta_-=d,$$

so they are a shadow pair. In standard quantization, the larger root is usually the CFT operator dimension. In the allowed alternative-quantization window, the smaller root can also define a consistent boundary operator.

Takeaway

A conformal family is the complete tower generated from one primary by translations:

$$[\mathcal O] = \left\{\mathcal O,\partial\mathcal O,\partial^2\mathcal O,\ldots\right\}.$$

The quadratic Casimir labels the entire family:

$$C_{\Delta,\rho} = \Delta(\Delta-d)+C_2^{SO(d)}(\rho),$$

and for symmetric traceless spin $\ell$,

$$C_{\Delta,\ell} = \Delta(\Delta-d)+\ell(\ell+d-2).$$

The OPE sums over families, and a conformal block is the contribution of one family to a four-point function. The Casimir equation is the bridge between representation theory and practical calculations of conformal blocks. In AdS/CFT, the same Casimir is the group-theoretic label of the corresponding bulk representation.