The Embedding-Space Formalism

Embedding space realizes the $d$-dimensional conformal group as a linear orthogonal group in $d+2$ dimensions. Physical points become projective null rays, tensor indices become polarization polynomials, and transversality removes components that vanish on pullback. The method is efficient only when its gauge redundancies, signature, homogeneity, and spin-cover conventions are kept explicit.

Required background. Spin and tensor representations provide the physical-space representations being lifted. Spinning tensor structures provide the correlator-counting problem that embedding space solves.

Helpful background. Lorentz, field, and Poincaré representations explain the distinction between tensor representations and their spin covers.

The projective null cone

For Euclidean physical space, use embedding space $\mathbb R^{d+1,1}$ with light-cone coordinates

$$P^A=(P^+,P^-,P^\mu)$$

and metric

$$P\cdot Q =-\frac12(P^+Q^-+P^-Q^+)+P^\mu Q_\mu.$$

Physical points are null rays

$$P^2=0, \qquad P\sim\lambda P, \qquad \lambda>0.$$

Choose the Poincaré section

$$P_x=(1,x^2,x^\mu).$$

Then

$$P_x^2=0, \qquad P_{ij}\equiv-2P_i\cdot P_j=x_{ij}^2.$$

A scalar primary of dimension $\Delta$ is represented by a homogeneous function

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

Its unit-normalized two-point function is simply

$$\langle\mathcal O(P_1)\mathcal O(P_2)\rangle =P_{12}^{-\Delta},$$

which pulls back to $|x_{12}|^{-2\Delta}$. The projective scaling is not an extra physical symmetry: it identifies different representatives of the same point.

Symmetric traceless tensors

Encode a rank-$\ell$ symmetric traceless tensor with an auxiliary vector $Z^A$:

$$\mathcal O(P,Z) =Z^{A_1}\cdots Z^{A_\ell}\mathcal O_{A_1\cdots A_\ell}(P).$$

Impose

$$Z^2=0, \qquad Z\cdot P=0,$$

and the equivalence

$$Z\sim Z+\beta P.$$

The complete homogeneity statement is

$$\mathcal O(\lambda P,\alpha Z+\beta P) =\lambda^{-\Delta}\alpha^\ell\mathcal O(P,Z).$$

Terms proportional to $P_A$ are pure gauge because they vanish when projected with $\partial P_x^A/\partial x^\mu$. Terms proportional to $Z^2$ encode traces and can be dropped in the null-polarization polynomial. These are quotient statements; setting an unconstrained $Z$ to zero would discard physical components.

For a physical null polarization $z^2=0$, take

$$Z_{x,z}=(0,2x\cdot z,z^\mu).$$

It satisfies $Z_{x,z}\cdot P_x=0$ and $Z_{x,z}^2=z^2=0$. The pullback is

$$\mathcal O(x,z)=\mathcal O(P_x,Z_{x,z}).$$

The basic gauge-invariant antisymmetric tensor is

$$C_i^{AB}=Z_i^AP_i^B-Z_i^BP_i^A.$$

Two useful invariants are

$$H_{ij}=-C_i\cdot C_j =-2\left[(Z_i\cdot Z_j)(P_i\cdot P_j) -(Z_i\cdot P_j)(P_i\cdot Z_j)\right],$$ $$V_{i,jk}=\frac{P_j\cdot C_i\cdot P_k}{P_j\cdot P_k}.$$

They are invariant under $Z_i\to Z_i+\beta_iP_i$. Costa, Penedones, Poland, and Rychkov derive these structures and their physical pullbacks in Costa et al. 2011, §§2–4, pp. 3–17.

An exact tensor three-point structure

For two scalars and one spin-$\ell$ symmetric-traceless primary, conformal covariance leaves one parity-even structure. Define

$$N_{3,12} =(Z_3\cdot P_1)(P_2\cdot P_3) -(Z_3\cdot P_2)(P_1\cdot P_3).$$

Then

$$\begin{aligned} &\langle\mathcal O_1(P_1)\mathcal O_2(P_2) \mathcal O_{3,\ell}(P_3,Z_3)\rangle\\ &\quad=\lambda_{123} \frac{N_{3,12}^{\ell}} {P_{12}^{(\Delta_1+\Delta_2-\Delta_3+\ell)/2} P_{23}^{(\Delta_2+\Delta_3-\Delta_1+\ell)/2} P_{31}^{(\Delta_3+\Delta_1-\Delta_2+\ell)/2}}. \end{aligned}$$

Every exponent follows from projective homogeneity. Substituting $P_i=P_{x_i}$ and $Z_3=Z_{x_3,z_3}$ recovers the physical tensor polynomial. For identical scalar operators, exchanging $1\leftrightarrow2$ changes $N_{3,12}^{\ell}$ by $(-1)^\ell$, so Bose symmetry sets the coefficient to zero for odd $\ell$.

This gives two independent checks: the pullback must have the correct physical scaling at each point, and the exchange sign must match the statistics of the external fields.

A concrete spinor lift in three dimensions

Spinors require the double cover and are dimension-specific. As a concrete Lorentzian three-dimensional convention, use

$$\operatorname{Spin}(3,2)\simeq\operatorname{Sp}(4,\mathbb R)$$

and a commuting auxiliary spinor $S_I$. Let

$$X_{IJ}=X^A(\Gamma_A)_{IJ}, \qquad S_IX^{IJ}=0.$$

An embedding spinor polynomial obeys

$$\Psi(aX,bS)=a^{-\Delta_\psi-1/2}b\,\Psi(X,S).$$

Equivalently, the field representative has the gauge freedom

$$\Psi^I(X)\sim\Psi^I(X)+X^{IJ}\Theta_J(X).$$

On the Poincaré section, a physical two-component polarization $s_\alpha$ is lifted by a section-dependent intertwiner $S_I(X,s)$ satisfying $SX=0$. With the conventions of Iliesiu et al. 2016, §2.2, pp. 5–8, the embedding two-point structure pulls back as

$$\frac{\langle S_1S_2\rangle}{X_{12}^{\Delta_\psi+1/2}} \longmapsto \frac{s_1^\alpha(x_{12})_{\alpha\beta}s_2^\beta} {(x_{12}^2)^{\Delta_\psi+1/2}}.$$

For two equal-dimension spinors and a scalar of dimension $\Delta_3$, one parity-even embedding structure is

$$\langle\Psi_1\Psi_2\mathcal O_3\rangle =\lambda \frac{\langle S_1S_2\rangle} {X_{12}^{\Delta_\psi-\Delta_3/2+1/2} X_{23}^{\Delta_3/2}X_{31}^{\Delta_3/2}}.$$

Its physical pullback is

$$\langle\psi(x_1,s_1)\psi(x_2,s_2)\mathcal O(x_3)\rangle =\widetilde\lambda \frac{s_1^\alpha(x_{12})_{\alpha\beta}s_2^\beta} {(x_{12}^2)^{\Delta_\psi-\Delta_3/2+1/2} (x_{23}^2x_{31}^2)^{\Delta_3/2}}.$$

The phase relating $\lambda$ and $\widetilde\lambda$ depends on Majorana and Wick-rotation conventions. Additional parity-odd structures can exist; Fierz identities and identical-fermion antisymmetry reduce the basis. A formula obtained in Lorentzian $\operatorname{Sp}(4,\mathbb R)$ conventions cannot be imported into Euclidean signature without specifying the analytic continuation and reality condition.

From physical data to blocks

The figure below organizes the tensor-polarization lift, invariant construction, and pullback. It displays the two tensor quotient steps: $P\sim\lambda P$ removes the choice of ray representative, while $Z\sim Z+\beta P$ removes unphysical polarizations. The corresponding spinor realization is recorded in the structured table below.

The embedding-space pipeline linearizes conformal covariance while preserving projective homogeneity, polarization gauge equivalence, spin-cover data, and physical-section checks. The diagram is schematic and not to scale.

The same pipeline is given in structured form:

Stage	Tensor realization	Three-dimensional spinor realization	Required check
Physical operator	$\mathcal O_{\mu_1\ldots\mu_\ell}(x)$	$\psi_\alpha(x)$ or a symmetric spinor	Physical representation and reality condition
Lift	Null ray $P^2=0$, homogeneous tensor	Null $X$ with $\operatorname{Sp}(4,\mathbb R)$ spinor	Embedding signature and spin cover
Polarization	$Z^2=Z\cdot P=0$, $Z\sim Z+\beta P$	$SX=0$ or the equivalent field gauge shift	No gauge-dependent numerator
Invariants	$C_i$, $H_{ij}$, $V_{i,jk}$	Spinor brackets and gamma-matrix strings	Homogeneity and Fierz identities
Pullback	$P_x$, $Z_{x,z}$	Section intertwiner $S(X,s)$	Correct physical scaling, traces, and exchange signs
Block input	A basis of three-point structures	Parity- and statistics-resolved structures	Basis independence of the final correlator

Failure tests

• Rescale one $P_i$ and verify the correlator changes only by its declared homogeneity.
• Shift one $Z_i\to Z_i+\beta P_i$ and verify exact invariance modulo $P_i^2=0$.
• Pull back a scalar two-point function and one spinning structure; recover $P_{ij}=x_{ij}^2$ and the expected inversion tensor.
• Count structures again after fixing the integer dimension. Gram and epsilon-tensor identities can make a generic-$d$ basis redundant.
• For spinors, check the spin cover, Fierz identities, chirality or Majorana condition, and statistics phase before imposing crossing.

The generation of conformal blocks from these structures continues on Spinning operators and blocks in higher dimensions.

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