Conformal Blocks and Casimir Equations

A conformal block is the contribution of one irreducible conformal family to a correlator in one OPE channel and one normalization. The quadratic Casimir turns this representation-theoretic definition into a differential equation. The equation alone is insufficient because the dimensions $\Delta$ and $d-\Delta$ have the same eigenvalue; the physical block is selected by its OPE asymptotic, channel, and branch.

Required background. From the Local OPE to Conformal Data supplies primary and descendant data. Cross Ratios and Four-Point Kinematics fixes $u,v,z,\bar z$ and channel maps. Helpful background. Descendant States and Gram Matrices supplies the level-by-level family sum.

One family in a four-point function

For four identical Hermitian scalars,

$$\langle\phi_1\phi_2\phi_3\phi_4\rangle =\frac{\mathcal G(u,v)} {(x_{12}^2x_{34}^2)^{\Delta_\phi}},$$

and in the $(12)(34)$ channel,

$$\mathcal G(u,v) =\sum_{\mathcal O} \lambda_{\phi\phi\mathcal O}^{\,2} g_{\Delta,\ell}(u,v).$$

This defines $g_{\Delta,\ell}$ after the leading OPE normalization is fixed. For $d>2$, with

$$\eta=\frac{1-v}{2\sqrt u}$$

held finite as $u\to0$, choose

$$\boxed{ g_{\Delta,\ell}(u,v) \sim u^{\Delta/2} \frac{C_\ell^{(\nu)}(\eta)} {C_\ell^{(\nu)}(1)}, \qquad \nu=\frac{d-2}{2}. }$$

The normalization makes the leading collinear Gegenbauer polynomial equal to one at $\eta=1$. At $d=2$, the $\nu\to0$ limit is reorganized into left- and right-moving structures; at $d=1$, there is no transverse angle and the separate one-dimensional block convention applies. Other sources may absorb $2^\ell$, $(-1)^\ell$, or $4^\Delta$ into the block or OPE coefficient. Such blocks solve the same equation but cannot be mixed in one crossing sum.

For nonidentical or spinning external operators, blocks carry external-dimension differences and tensor-structure indices. A block matrix connects one left three-point structure to one right structure; it is not a new dynamical coefficient.

The quadratic Casimir equation

Let $J_{AB}^{(i)}$ act on insertion $i$ and define

$$\mathcal C_{12} =\frac12 \left(J_{AB}^{(1)}+J_{AB}^{(2)}\right) \left(J^{AB(1)}+J^{AB(2)}\right).$$

An exchanged symmetric traceless primary of dimension $\Delta$ and spin $\ell$ has eigenvalue

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

Therefore the family contribution obeys

$$\boxed{ \left(\mathcal C_{12}-C_{\Delta,\ell}\right) \left[\text{prefactor}\times g_{\Delta,\ell}(u,v)\right]=0. }$$

After stripping the prefactor, this becomes a second-order equation in $z,\bar z$. One widely used scalar convention sets

$$a=-\frac{\Delta_{12}}2, \qquad b=\frac{\Delta_{34}}2,$$

and

$$\begin{aligned} \mathcal D={}&D_z+D_{\bar z} +2\nu\frac{z\bar z}{z-\bar z} \left[(1-z)\partial_z-(1-\bar z)\partial_{\bar z}\right],\\ D_z={}&z^2(1-z)\partial_z^2 -(a+b+1)z^2\partial_z-abz. \end{aligned}$$

Then

$$\mathcal D\,g_{\Delta,\ell}^{a,b} =\frac12C_{\Delta,\ell}g_{\Delta,\ell}^{a,b}.$$

The apparent pole at $z=\bar z$ is removable on solutions with the correct exchange symmetry. It is a coordinate artifact of reducing the two-variable equation, not a physical singularity of a regular Euclidean correlator. Dolan and Osborn derive the Casimir system and closed forms in special even dimensions in Dolan and Osborn 2004, §§ 2–3.

Selecting the block rather than its shadow

Because

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

the same differential equation admits the shadow asymptotic $u^{(d-\Delta)/2}$. The OPE boundary condition selects $u^{\Delta/2}$. At $\Delta=d/2$ the two indicial roots meet and logarithmic solutions can appear; the physical prescription is obtained by continuation with a declared normalization.

Additional checks are needed:

Check	Physical block	Wrong or incomplete solution
Channel	Singular behavior tied to $x_1\to x_2$	An eigenfunction normalized in another pairing
Leading power	$u^{\Delta/2}$	$u^{(d-\Delta)/2}$ shadow behavior
Spin	Gegenbauer degree $\ell$ and correct exchange parity	A solution with another angular representation
Regularity	Euclidean regularity away from OPE singularities	A spurious singular branch
Null quotient	Shortening removes null descendants	A long block evaluated at a short value without subtraction
External convention	Same $a,b$ and prefactor as the correlator	Correct function multiplied by inconsistent powers

At shortening, the irreducible block can be obtained by subtracting the null-family block or by taking a regulated limit. A naive substitution into a long-block formula can leave a pole whose residue is precisely the null submodule.

Radial expansion and checks

Let $\rho=re^{i\theta}$ and $\eta=\cos\theta$ in the symmetric radial frame. In the standard $z$-normalized convention, $z\sim4\rho$ near the OPE limit, so

$$g_{\Delta,\ell}(r,\eta) =(4r)^\Delta \sum_{n=0}^{\infty}r^n \sum_j B_{n,j}\, \frac{C_j^{(\nu)}(\eta)}{C_j^{(\nu)}(1)}.$$

The leading term in brackets starts with the spin-$\ell$ primary. Descendant levels determine the allowed $j$ and coefficients. For identical Hermitian external scalars in a reflection-positive theory, a compatible radial normalization gives nonnegative contributions to the appropriate radial matrix element, providing useful coefficient checks. This statement is not a claim that every coefficient in every tensor basis is positive.

The rapid convergence of the $\rho$ series and its descendant interpretation are developed in Hogervorst and Rychkov 2013, §§ 2–4.

From data to blocks to crossing

The governed flow figure emphasizes that a block sits between primary data and a channel sum.

The Casimir eigenvalue fixes a differential equation, while the OPE asymptotic chooses the physical block over the shadow solution. Blocks are kinematic functions; OPE tensors supply dynamical weights. A convergent sum reconstructs one channel, associativity equates channels, and reflection positivity supplies nonnegative scalar or positive-semidefinite matrix weights. The diagram is schematic.

Input	Operation	Result	Independent check
$(\Delta,\ell)$	Evaluate $C_{\Delta,\ell}$	Casimir eigenvalue	Invariant under $\Delta\leftrightarrow d-\Delta$
External dimensions and prefactor	Form $\mathcal D^{a,b}$	Differential equation	Correct transformation under point permutations
OPE boundary data	Select $u^{\Delta/2}C_\ell^{(\nu)}$	Physical channel block	Excludes the shadow root
Descendant module	Expand in $\rho$	Radial coefficients	Level and spin match the Gram decomposition
Short module	Remove null family	Irreducible short block	Null equation and recombination limit agree
Spectrum and OPE tensors	Sum blocks	Channel correlator	Converges in the declared radial domain

Conformal Partial Waves and the Shadow Formalism constructs the block-plus-shadow solution by an integral. Crossing Equations and Positivity uses the normalized block in a positive sum rule. Release-pinned block generation and numerical benchmarks remain outside this page.

Common pitfalls

The Casimir eigenvalue uniquely determines a block. It is invariant under $\Delta\leftrightarrow d-\Delta$. The OPE boundary condition and branch select the physical solution.

A block contains an OPE coefficient. The block is kinematic once external and exchanged representations are fixed. OPE tensors multiply it.

Short blocks follow by blind substitution. Null descendants can produce poles or reducible contributions. Take the irreducible quotient or a controlled recombination limit.

Exercises

Verify the shadow degeneracy of the quadratic Casimir.

Solution

Substitution gives

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

The spin term is unchanged, so $C_{d-\Delta,\ell}=C_{\Delta,\ell}$.

References

• Dolan, F. A., and Hugh Osborn. “Conformal Partial Waves and the Operator Product Expansion.” Nuclear Physics B 678 (2004): 491–507. DOI; Open PDF
• Hogervorst, Matthijs, and Slava Rychkov. “Radial Coordinates for Conformal Blocks.” Physical Review D 87 (2013): 106004. DOI; Open PDF