Spinning Operators and Blocks in Higher Dimensions

A spinning conformal block is a matrix-valued change of basis between pairs of three-point tensor structures. It is not specified by $(\Delta,\ell)$ alone: external representations, parity and chirality sectors, statistics phases, block normalization, and the bases at both OPE vertices are part of its definition. Differential and weight-shifting operators make these blocks calculable from a small set of seeds.

Required background. The embedding-space formalism fixes projective and polarization conventions. Conformal blocks and Casimir equations provide the scalar eigenvalue problem and OPE boundary conditions.

Helpful background. Partial waves and the shadow formalism explain the integral construction and shadow subtraction used for many seed blocks.

A block carries two structure labels

Choose bases $t_{12\mathcal O}^{(a)}$ and $t_{34\mathcal O}^{(b)}$ for the three-point functions at the two OPE vertices. A four-point tensor component then has the decomposition

$$\mathcal G_I(u,v) =\sum_{\mathcal O}\sum_{a,b} \lambda_{12\mathcal O}^{(a)} \lambda_{34\mathcal O}^{(b)} G_{\mathcal O,I}^{ab}(u,v).$$

The label $I$ selects a four-point tensor structure, while $a,b$ select the left and right three-point structures. If a three-point basis changes by $t'^{(a)}=M^a{}_c t^{(c)}$, then the OPE coefficients transform by $M^{-1}$ and the block matrix transforms oppositely. Only the complete contraction is basis independent.

For exchange of a symmetric-traceless primary of dimension $\Delta$ and spin $\ell$, the quadratic conformal Casimir eigenvalue is

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

The coupled block components obey

$$\frac12\left(\mathcal L_1+\mathcal L_2\right)_{AB} \left(\mathcal L_1+\mathcal L_2\right)^{AB} G_{\Delta,\ell,I}^{ab} =C_{\Delta,\ell}G_{\Delta,\ell,I}^{ab},$$

where $\mathcal L_i$ contains both orbital and spin generators. The Casimir equation alone also admits the shadow solution with dimension $d-\Delta$; the OPE boundary condition selects the physical block.

We normalize the block by the leading radial OPE contribution of a unit-normalized exchanged primary. In radial variables $(r,\eta)$,

$$G_{\Delta,\ell,I}^{ab}(r,\eta) =r^\Delta\left[T_{\ell,I}^{ab}(\eta)+O(r)\right].$$

The tensor polynomial $T_{\ell,I}^{ab}$ is part of the convention. Comparing two codes through only the coefficient of $r^\Delta$ is insufficient if their three-point bases differ.

Seed blocks and differential generation

A seed block is a block with the smallest external representations compatible with the exchanged representation and with one structure at each vertex. More general blocks can be generated schematically as

$$G_{\Delta,\rho}^{ab} =\mathcal D_{12}^{(a)}\mathcal D_{34}^{(b)} G_{\Delta',\rho'}^{\mathrm{seed}},$$

where the external dimensions and representations of the seed may be shifted. The differential operators must be moved through the OPE prefactors consistently; applying them only to the reduced function generally gives the wrong normalization.

Weight-shifting operators provide a representation-theoretic version of this construction. A covariant operator

$$\mathcal D_m: [\Delta,\rho] \longrightarrow[\Delta+\delta\Delta_m,\rho_m]$$

changes dimension and spin while carrying a finite-dimensional conformal representation. Moving $\mathcal D_m$ from one leg of a three-point function to another gives a finite crossing relation whose coefficients are conformal $6j$ symbols. Repeated moves reduce a general external representation to scalar or minimal fermionic seeds. The construction and its normalization factors are derived in Karateev, Kravchuk, and Simmons-Duffin 2018, §§2–4, pp. 5–50.

The method does not mean that every block is one fixed derivative of a scalar block. There can be several seeds, zeros or poles in the change-of-basis matrices, and dimension-specific relations among structures. The representation shifts and the external prefactors must be recorded with the result.

A conserved-current block workflow

Consider four identical conserved currents in a parity-even sector. In $d$ dimensions,

$$\Delta_J=d-1, \qquad \partial^\mu J_\mu=0.$$

A practical construction is:

1. enumerate the $JJ\mathcal O$ three-point structures in embedding space;
2. generate unconstrained blocks from scalar or minimal spinning seeds;
3. apply the physical divergence at points $1$ and $2$;
4. solve the resulting linear relations among the $JJ\mathcal O$ structures;
5. normalize the surviving combination by its leading OPE tensor;
6. verify the Casimir equation and the identical-current exchange parity.

Away from coincident insertions, conservation requires

$$\sum_a c_a\, \partial_{x_1}\cdot G_{\Delta,\rho,I}^{ab,(a)}(x_i)=0.$$

At coincidence, Ward identities permit contact terms, so a separated-point block must not be used to determine them. For stress tensors the corresponding external dimension is $\Delta_T=d$, and trace/improvement conventions must be fixed before imposing conservation.

The embedding-space divergence operator and the counting of conserved structures are given in Costa et al. 2011, §§5–6, pp. 28–38.

Fermion–scalar seeds and statistics

In three dimensions, a fermion–scalar seed can exchange half-integer spin. The seed carries an explicit spinor structure, and weight shifting relates it to scalar blocks through a finite-order differential operator. The external fermion exchange acts on both coordinates and structure labels:

$$\mathcal G_I(1,2,3,4) =-\mathsf P_I{}^J \mathcal G_J(2,1,3,4),$$

where the minus sign is Grassmann statistics and $\mathsf P$ is the tensor-basis permutation matrix. Omitting either factor can place an exchanged operator in the wrong parity or spin sector.

For a free fermion, $\Delta_\psi=(d-1)/2$ and the Dirac descendant is null, so

$$\gamma^\mu\partial_\mu\psi=0$$

must annihilate the corresponding external block away from contacts. A generic interacting fermion has $\Delta_\psi>(d-1)/2$ and does not satisfy this shortening condition. Imposing it merely because the external representation is spinorial deletes physical blocks.

In three dimensions, identical Majorana-fermion three-point structures separate into parity-even and parity-odd sectors and obey additional exchange constraints. Iliesiu and collaborators construct the structures and express their blocks through derivatives of scalar blocks in Iliesiu et al. 2016, §§2.4–2.5, pp. 9–17.

Recursion and stability

Conformal blocks are meromorphic functions of $\Delta$. At special values, descendants become primary and generate poles whose residues are lower-spin or shifted-dimension blocks. A recursion has the schematic form

$$G_{\Delta,\rho} =G_{\infty,\rho} +\sum_A\frac{R_A}{\Delta-\Delta_A^*} G_{\Delta_A',\rho_A'}.$$

For spinning blocks, $R_A$ is a matrix in structure space. Numerical stability requires:

• separating physical poles from basis singularities;
• evaluating near-degenerate residues at sufficient precision;
• checking recursion depth and radial-series order independently;
• verifying the Casimir residual and leading OPE tensor after basis conversion;
• testing a shortened or exactly known limit.

Apparent blow-up of individual components can cancel in the physical structure combination. Rescaling or orthogonalizing the tensor basis may be necessary before diagnosing a failed recursion.

The spinning-correlator pipeline

The figure below connects the representation data to a crossing-ready block. Inspect the polarization-gauge test and the final physical pullback. The adjacent table supplies the statistics, shortening, Casimir, and OPE-normalization checks that must accompany the displayed transformations.

Spinning blocks use a declared embedding-space tensor basis and are accepted only after Casimir, OPE, statistics, parity or chirality, shortening, and physical-pullback checks. The diagram is schematic and not to scale.

The pipeline has this structured equivalent:

Input or step	Required data	Output	Failure test
External operators	Dimension, tensor or spinor representation, reality, parity, chirality	Allowed three-point bases	Recount in the target integer dimension
Exchanged operator	$\Delta$, representation, conservation status	Casimir eigenvalue and OPE tensor	Compare to unitarity and shortening conditions
Seed choice	Minimal structures and block normalization	Scalar or fermion–scalar seed	Verify leading radial term and shadow removal
Differential generation	Every dimension/spin shift and prefactor	Unconstrained spinning blocks	Apply the Casimir operator after generation
Physical constraints	Conservation, Dirac shortening when applicable, traces	Surviving linear combinations	Check separated-point divergences
Crossing conversion	Permutation matrices and fermion signs	Crossing-ready vector of blocks	Apply the exchange twice and recover the original basis
Numerical recursion	Pole data, precision, radial order	Evaluated blocks	Vary recursion depth and inspect residuals

For the Ward-normalized conserved coefficients that multiply these blocks, continue to Current and stress-tensor CFT data.

Common pitfalls

A differential block is not normalized automatically. Weight shifting changes external dimensions and three-point bases. Restore the declared OPE leading term before comparing OPE coefficients.

Conservation is not a generic spin condition. It occurs only at a shortening dimension and can have contact terms. Do not impose it on an unshortened exchanged or external representation.

Parity and chirality are dimension-specific. Epsilon-tensor identities, Weyl projections, and Majorana conditions can change the number and reality of structures.

References

• Costa, Miguel S., João Penedones, David Poland, and Slava Rychkov. “Spinning Conformal Correlators.” Journal of High Energy Physics 2011, no. 11 (2011): 071. DOI. Open PDF
• Iliesiu, Luca, Filip Kos, David Poland, Silviu S. Pufu, David Simmons-Duffin, and Ran Yacoby. “Bootstrapping 3D Fermions.” Journal of High Energy Physics 2016, no. 3 (2016): 120. DOI. Open PDF
• Karateev, Denis, Petr Kravchuk, and David Simmons-Duffin. “Weight Shifting Operators and Conformal Blocks.” Journal of High Energy Physics 2018, no. 2 (2018): 081. DOI. Open PDF