Spinor Conformal Primary Wavefunctions

• Spinor conformal primary wavefunctions are solutions to the Dirac equation that, in addition to satisfying on‑shell conditions, transform under SL(2,C) as two-dimensional conformal primaries with specific conformal dimensions.
• They utilize techniques such as the Mellin transform, spin-frame methods, and shadow transforms to bridge momentum space with celestial correlators and provide a complete scattering basis.
• Both massless and massive cases are constructed, enabling the reformulation of scattering amplitudes into conformally covariant correlators that underpin celestial holography.

Searching arXiv for recent and foundational papers on spinor conformal primary wavefunctions. Spinor conformal primary wavefunctions are solutions of relativistic spinor field equations in flat spacetime that are labeled not by on-shell momentum alone, but by a point on the celestial sphere and a conformal dimension, and that transform under the Lorentz group as conformal primaries on that sphere. In four-dimensional Minkowski space, this reformulates the action of $SO(1,3)\simeq SL(2,\mathbb{C})$ as a two-dimensional conformal action on celestial coordinates, so that scattering amplitudes with spinorial external states can be rewritten as conformally covariant correlators. The subject developed from the general conformal-primary basis for flat-space amplitudes (Pasterski et al., 2017), acquired explicit fermionic realizations for Dirac spinors and massive celestial fermions (Iacobacci et al., 2020, Narayanan, 2020), and was further organized through spin-frame constructions, shadow transforms, generalized primaries, and conformal multiplet structure in celestial CFT (Pasterski et al., 2020, Pasterski et al., 2021). A more group-theoretic formulation based on an emergent $sl(2,\mathbb{C})\times D$ symmetry on the solution space of massless fields provides explicit spinor constructions and Casimir constraints (Chen et al., 2023).

1. Lorentz symmetry, celestial coordinates, and the definition of spinor primaries

The basic geometric datum is the celestial sphere, whose points label null directions. In four-dimensional Minkowski space with metric

$\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1),$

a null direction can be written in complex coordinates $(z,\bar z)$ through

$$q^\mu (z,\bar z)=(1+|z|^2,\; z+\bar z,\; -i(z-\bar z),\; 1-|z|^2),$$

which transforms under a Lorentz transformation $\Lambda\in SL(2,\mathbb{C})$ by a Möbius map

$$z\to z'=\frac{az+b}{cz+d},\qquad \bar z\to \bar z'=\frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$

together with an overall conformal factor (Pasterski et al., 2017). This is the kinematic origin of celestial conformal covariance.

A conformal primary wavefunction is defined by two simultaneous requirements. First, it must solve the relevant free field equation, such as the massless Dirac equation or the massive Dirac equation. Second, under Lorentz transformations it must transform as a two-dimensional conformal primary on the celestial sphere, with definite conformal weights. In the general massless-field formulation, a wavefunction $\mathcal O(x;w,\bar w)$ is called a conformal primary of Lorentz dimension $\Delta$ and two-dimensional spin $J$ if under the induced Möbius transformation it obeys

$sl(2,\mathbb{C})\times D$0

where $sl(2,\mathbb{C})\times D$1 acts on spinor or tensor indices (Chen et al., 2023). For spin-$sl(2,\mathbb{C})\times D$2 fields, the spin label is $sl(2,\mathbb{C})\times D$3, and the corresponding two-dimensional weights are

$sl(2,\mathbb{C})\times D$4

In four dimensions, the spinor-helicity variables adapted to the celestial parameterization are fixed by the null momentum $sl(2,\mathbb{C})\times D$5, with $sl(2,\mathbb{C})\times D$6, and may be written as

$sl(2,\mathbb{C})\times D$7

up to in/out sign conventions (Pasterski et al., 2017). This directly ties spinor indices to the Möbius action on $sl(2,\mathbb{C})\times D$8, and makes the spinorial generalization of scalar and vector conformal primaries natural.

A complementary arbitrary-dimensional definition was given for Dirac spinors in $sl(2,\mathbb{C})\times D$9. There, spinor conformal primary wavefunctions are solutions to the Dirac equation that are parameterized by a point $\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1),$0 and a conformal dimension $\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1),$1, and transform as $\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1),$2-dimensional conformal primary spinors under $\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1),$3 (Iacobacci et al., 2020). This places the four-dimensional celestial construction inside a broader representation-theoretic framework.

2. Explicit constructions for massless and massive spinors

For massless spinors in four dimensions, the simplest radiative conformal primaries are built from a scalar conformal factor and a spin frame. Using

$\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1),$4

the left-handed radiative spin-$\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1),$5 primaries are

$\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1),$6

where $\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1),$7 and $\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1),$8 are spin-frame elements adapted to $\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1),$9 and the bulk point $(z,\bar z)$0 (Pasterski et al., 2021). In the spin-frame formalism, $(z,\bar z)$1 has $(z,\bar z)$2 and $(z,\bar z)$3 has $(z,\bar z)$4, while the scalar factor carries $(z,\bar z)$5 but no spin (Pasterski et al., 2020).

A more explicit group-theoretic construction introduces a bulk-dependent spinor frame

$(z,\bar z)$6

where $(z,\bar z)$7 encodes the null direction and $(z,\bar z)$8 is the bulk dilatation vector (Chen et al., 2023). In this language, the elementary spinor conformal primary

$(z,\bar z)$9

has

$$q^\mu (z,\bar z)=(1+|z|^2,\; z+\bar z,\; -i(z-\bar z),\; 1-|z|^2),$$0

satisfies the massless Weyl equation, and is an explicit conformal primary (Chen et al., 2023). More generally, for negative helicity $$q^\mu (z,\bar z)=(1+|z|^2,\; z+\bar z,\; -i(z-\bar z),\; 1-|z|^2),$$1 the standard family is

$$q^\mu (z,\bar z)=(1+|z|^2,\; z+\bar z,\; -i(z-\bar z),\; 1-|z|^2),$$2

with shadow-like partners involving $$q^\mu (z,\bar z)=(1+|z|^2,\; z+\bar z,\; -i(z-\bar z),\; 1-|z|^2),$$3 and powers of $$q^\mu (z,\bar z)=(1+|z|^2,\; z+\bar z,\; -i(z-\bar z),\; 1-|z|^2),$$4 (Chen et al., 2023). For spin-$$q^\mu (z,\bar z)=(1+|z|^2,\; z+\bar z,\; -i(z-\bar z),\; 1-|z|^2),$$5, only the first and third families survive as independent solutions; the mixed family exists only for $$q^\mu (z,\bar z)=(1+|z|^2,\; z+\bar z,\; -i(z-\bar z),\; 1-|z|^2),$$6 (Chen et al., 2023).

For massive Dirac fermions in four dimensions, the conformal primary wavefunctions are built by integrating over the unit hyperboloid $$q^\mu (z,\bar z)=(1+|z|^2,\; z+\bar z,\; -i(z-\bar z),\; 1-|z|^2),$$7. A massive momentum is written as $$q^\mu (z,\bar z)=(1+|z|^2,\; z+\bar z,\; -i(z-\bar z),\; 1-|z|^2),$$8, with $$q^\mu (z,\bar z)=(1+|z|^2,\; z+\bar z,\; -i(z-\bar z),\; 1-|z|^2),$$9, and the wavefunctions take the bulk-to-boundary form

$\Lambda\in SL(2,\mathbb{C})$0

where $\Lambda\in SL(2,\mathbb{C})$1 is the scalar bulk-to-boundary propagator and $\Lambda\in SL(2,\mathbb{C})$2 is a 4-component spinor fixed by conformal covariance and the Dirac equation (Narayanan, 2020). Explicit formulas were derived for both $\Lambda\in SL(2,\mathbb{C})$3, and the shadow transform was shown to map

$\Lambda\in SL(2,\mathbb{C})$4

for four-dimensional celestial fermions (Narayanan, 2020).

In arbitrary dimension, the massive spinor conformal primary wavefunctions admit an embedding-space integral representation,

$\Lambda\in SL(2,\mathbb{C})$5

while the massless solutions arise as a Mellin transform over null energy,

$\Lambda\in SL(2,\mathbb{C})$6

(Iacobacci et al., 2020). This makes precise the statement that, in the massless case, the conformal primary basis is related to momentum space by a Mellin transform.

3. Mellin transform, principal series, and completeness

The change of basis from momentum eigenstates to conformal primaries is implemented by a Mellin transform in the energy for massless fields. In the general flat-space conformal basis, the massless scalar conformal primary is

$\Lambda\in SL(2,\mathbb{C})$7

and the same Mellin logic extends to spinning fields because the spin dependence resides in $\Lambda\in SL(2,\mathbb{C})$8-independent polarizations (Pasterski et al., 2017). For spinors, the massless conformal primary basis is therefore obtained by Mellin transforming plane-wave spinors with null momentum $\Lambda\in SL(2,\mathbb{C})$9 (Pasterski et al., 2017, Iacobacci et al., 2020).

The dimensions that yield delta-function-normalizable bases lie on the principal continuous series. For general massless fields in $$z\to z'=\frac{az+b}{cz+d},\qquad \bar z\to \bar z'=\frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$0, the relevant locus is

$$z\to z'=\frac{az+b}{cz+d},\qquad \bar z\to \bar z'=\frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$1

In four dimensions this becomes

$$z\to z'=\frac{az+b}{cz+d},\qquad \bar z\to \bar z'=\frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$2

For massless fields, including spinors, this is the range required by Mellin unitarity and inner-product normalizability (Pasterski et al., 2017, Pasterski et al., 2017, Iacobacci et al., 2020). The massive case leads to the same principal-series condition, though for massive spinors one may choose either a single-spin basis with $$z\to z'=\frac{az+b}{cz+d},\qquad \bar z\to \bar z'=\frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$3, or a dual-spin basis with $$z\to z'=\frac{az+b}{cz+d},\qquad \bar z\to \bar z'=\frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$4 and $$z\to z'=\frac{az+b}{cz+d},\qquad \bar z\to \bar z'=\frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$5 (Narayanan, 2020).

The inner-product structure makes completeness explicit. In the arbitrary-dimensional Dirac case, the massless conformal primaries satisfy

$$z\to z'=\frac{az+b}{cz+d},\qquad \bar z\to \bar z'=\frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$6

while the massless plane-wave spinor may be recovered by the inverse Mellin transform

$$z\to z'=\frac{az+b}{cz+d},\qquad \bar z\to \bar z'=\frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$7

(Iacobacci et al., 2020). In four-dimensional massive celestial fermions, the Dirac inner product fixes the principal series $$z\to z'=\frac{az+b}{cz+d},\qquad \bar z\to \bar z'=\frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$8, and the resulting wavefunctions are delta-function normalizable (Narayanan, 2020). This establishes the conformal primary basis as a bona fide Hilbert-space basis rather than a merely formal reparameterization.

A group-theoretic reinterpretation was proposed in terms of an emergent $$z\to z'=\frac{az+b}{cz+d},\qquad \bar z\to \bar z'=\frac{\bar a\bar z+\bar b}{\bar c\bar z+\bar d},$$9 symmetry on the solution space of massless fields. There, novel Casimirs

$\mathcal O(x;w,\bar w)$0

act on the spinor solution space and relate the celestial weights $\mathcal O(x;w,\bar w)$1 to the bulk dilatation operator $\mathcal O(x;w,\bar w)$2 (Chen et al., 2023). This approach suggests a discrete basis built from highest- or lowest-weight primaries and descendants, in contrast with the usual continuous principal-series basis. The paper does not prove completeness of this discrete basis, and explicitly notes that completeness is not established there (Chen et al., 2023). This marks an active conceptual distinction rather than a contradiction: the Mellin/principal-series basis is a complete scattering basis, while the $\mathcal O(x;w,\bar w)$3 analysis emphasizes algebraic organization and new on-shell solutions.

4. Shadow transform, radiative versus generalized primaries, and spin-frame methods

The shadow transform is central to the structure of spinor conformal primaries. In arbitrary dimensions, the spinor shadow satisfies

$\mathcal O(x;w,\bar w)$4

so the shadow maps $\mathcal O(x;w,\bar w)$5 while preserving the bulk Dirac equation (Iacobacci et al., 2020). In four-dimensional celestial language, the shadow of a spinor primary flips the two-dimensional spin,

$\mathcal O(x;w,\bar w)$6

and for massive celestial fermions this was shown explicitly for Dirac wavefunctions (Narayanan, 2020).

For massless spinors in the spin-frame formalism, the shadow of the radiative spin-$\mathcal O(x;w,\bar w)$7 primary is

$\mathcal O(x;w,\bar w)$8

where $\mathcal O(x;w,\bar w)$9 and $\Delta$0 are the complementary spin-frame elements (Pasterski et al., 2021). This realizes the shadow not merely as an abstract integral transform but as another explicit bulk wavefunction with opposite celestial spin.

A useful distinction is between radiative conformal primary wavefunctions and generalized conformal primary wavefunctions. Radiative primaries are on-shell solutions of the relevant free equations and have $\Delta$1 for a spin-$\Delta$2 field. Generalized primaries retain the correct $\Delta$3 transformation law but need not satisfy the same on-shell or gauge constraints (Pasterski et al., 2021). For spin-$\Delta$4, inserting the generalized ansatz

$\Delta$5

into the Weyl equation shows that enforcing the equations of motion for all $\Delta$6 yields only the familiar radiative primaries and their shadows; there are no additional discrete generalized spin-$\Delta$7 solutions analogous to the vector and gravitino cases (Pasterski et al., 2021). This sharply distinguishes fermions of spin $\Delta$8 from higher-spin gauge fields.

The spin-frame formulation is especially powerful because it systematizes all radiative primaries up to spin 2. In this language, the spin-$\Delta$9 conformal primary is simply

$J$0

and the higher-spin primaries are obtained by multiplying by additional $J$1, $J$2, or spinor-frame factors (Pasterski et al., 2020). The same work showed how half-integer spin steps are related by supersymmetry, with the operator $J$3 acting as a spin-raising map from scalar to spinor primaries (Pasterski et al., 2020). This suggests a unifying representation-theoretic picture in which scalar, spinor, vector, and tensor conformal primaries occupy the same geometric ladder.

5. Celestial amplitudes and conformal correlators

Once external states are expanded in a conformal primary basis, scattering amplitudes transform as conformal correlators on the celestial sphere. For gluons, the Mellin transform of momentum-space amplitudes yields objects with the conformal covariance of two-dimensional correlators (Pasterski et al., 2017). The same statement extends to fermions: the arbitrary-dimensional Dirac conformal basis provides the external wavefunctions needed to rewrite fermionic scattering in a celestial basis (Iacobacci et al., 2020), while the four-dimensional massive construction gives the explicit transform from momentum-space amplitudes to celestial amplitudes for massive fermions (Narayanan, 2020).

For massive fermions, the celestial transform of an amplitude with external Dirac states is

$J$4

where the coefficients $J$5 express the conformal-primary spinor integrands in a conventional spin basis (Narayanan, 2020). This is the fermionic analog of the hyperbolic transform used for massive scalars and spinning bosons.

In the massless case, the transform reduces to Mellin integration in the energy of each external null momentum. This is the natural spinor generalization of the scalar Mellin transform emphasized in the original flat-space conformal-basis program (Pasterski et al., 2017). The Lorentz generators act as the global conformal generators of the celestial sphere, and the celestial amplitudes inherit the weights $J$6 of the spinor primaries (Iacobacci et al., 2020).

The two-dimensional conformal structure of spinorial correlators is not incidental. Embedding-space analyses of conformal correlators with spinors in general dimensions show that spinor two-point and three-point functions are fixed by homogeneity, transversality, and a small number of spinorial invariants (Isono, 2017). In particular, spinor correlators can often be written as differential operators acting on scalar correlators, and conformal blocks with spinors arise by acting on scalar conformal blocks with appropriate spinorial differential operators (Isono, 2017). Although this work is not phrased in celestial terms, it provides the natural CFT-side correlator structures expected from celestial amplitudes with spinor insertions.

A related recent flat-holographic construction for a free massive spinor in four-dimensional Minkowski space uses Milne slicing and $J$7 harmonics to derive a flat-space holographic dictionary. There, spinor conformal primary wavefunctions are realized as bulk solutions with near-$J$8-boundary behavior controlled by a celestial source, and the resulting two-point functions on the celestial sphere take the universal form dictated by two-dimensional conformal symmetry for spin-$J$9 primaries (Ageev et al., 6 Mar 2026). This suggests a close structural relation between the hyperbolic integral representation of massive celestial fermions (Narayanan, 2020) and a flat-space bulk-to-boundary map phrased in $sl(2,\mathbb{C})\times D$00-like terms (Ageev et al., 6 Mar 2026).

6. Multiplets, soft limits, and higher-spin extensions

The conformal primary basis also organizes spinor states into global conformal multiplets. In celestial CFT, global $sl(2,\mathbb{C})\times D$01 primary descendants at special values of $sl(2,\mathbb{C})\times D$02 form “celestial diamonds” (Pasterski et al., 2021). For spin-$sl(2,\mathbb{C})\times D$03, the structure is comparatively simple: there are radiative primaries and their shadows, but no nontrivial generalized primaries surviving all constraints. Accordingly, spin-$sl(2,\mathbb{C})\times D$04 exhibits only zero-area type-III diamond structures rather than the richer finite-area diamonds seen for gauge fields and gravitini (Pasterski et al., 2021).

At the conformally soft value $sl(2,\mathbb{C})\times D$05, the radiative spin-$sl(2,\mathbb{C})\times D$06 mode

$sl(2,\mathbb{C})\times D$07

gives a soft theorem but not a Goldstone mode in the same sense as large-gauge or large-supersymmetry Goldstones (Pasterski et al., 2021). Its shadow sits at $sl(2,\mathbb{C})\times D$08. The associated type-III descendant relation takes the form

$sl(2,\mathbb{C})\times D$09

with the conjugate relation for opposite chirality (Pasterski et al., 2021). This means the most subleading soft fermionic structures close under shadow rather than requiring additional generalized primaries.

The spin-$sl(2,\mathbb{C})\times D$10 extension is much richer and illustrates the utility of the spin-$sl(2,\mathbb{C})\times D$11 building blocks. Explicit conformal primary wavefunctions for the massless gravitino were constructed as

$sl(2,\mathbb{C})\times D$12

(Pasterski et al., 2021, Pasterski et al., 2020). Their conformally soft Goldstone mode occurs at $sl(2,\mathbb{C})\times D$13, where the spin-$sl(2,\mathbb{C})\times D$14 conformal primary becomes pure gauge and represents the Goldstone field for large local supersymmetry (Pasterski et al., 2020). In the massive setting, the Rarita–Schwinger conformal primary wavefunctions were constructed by combining massive spin-1 conformal primaries with Dirac conformal primaries using standard Clebsch–Gordan coefficients (Narayanan, 2020). This yields an explicit prescription for all massive half-integer spins once the spin-$sl(2,\mathbb{C})\times D$15 basis is known.

The same logic extends upward. Massive spin-1 conformal primaries can be built from Dirac conformal primary wavefunctions using standard Clebsch–Gordan coefficients, and the paper that developed massive celestial fermions used this to write massive spin-$sl(2,\mathbb{C})\times D$16 conformal primaries as well (Narayanan, 2020). This suggests that spin-$sl(2,\mathbb{C})\times D$17 is not merely one case among many, but the foundational fermionic building block for all higher-spin celestial wavefunctions.

A broader implication emerges from the group-theoretic $sl(2,\mathbb{C})\times D$18 analysis. For $sl(2,\mathbb{C})\times D$19, new on-shell conformal primary families appear beyond the familiar scalar-shadow pair, while for spin-$sl(2,\mathbb{C})\times D$20 the absence of the mixed family makes the fermionic sector especially rigid (Chen et al., 2023). This suggests that spinor conformal primary wavefunctions occupy a structurally minimal position within celestial representation theory: sufficiently rich to encode spinor scattering and shadow relations, but not rich enough to develop the same multiplicity of generalized modes that appear in gauge sectors.

Spinor conformal primary wavefunctions therefore lie at the intersection of several formalisms: Mellin transforms of plane waves, hyperbolic bulk-to-boundary integral kernels, spin-frame and null-tetrad geometry, embedding-space spinor representation theory, and celestial conformal multiplets. Across these approaches, the persistent elements are the same: a conformal dimension $sl(2,\mathbb{C})\times D$21, a celestial point $sl(2,\mathbb{C})\times D$22, a definite two-dimensional spin $sl(2,\mathbb{C})\times D$23, Lorentz covariance recast as conformal covariance, principal-series normalizability, and shadow relations linking dual descriptions of the same bulk solution. Together they provide the fermionic analog of the scalar, vector, and graviton celestial bases, and supply the natural language for fermionic scattering in celestial holography (Pasterski et al., 2017, Iacobacci et al., 2020, Narayanan, 2020).