Lectures on the Spinor and Twistor Formalism in 3D Conformal Field Theory (2508.21633v1)

Abstract: These notes are based on my lectures given at $\text{ST}^4$ 2025 held at IISER Bhopal. We study the application of spinor and twistor methods to three dimensional conformal field theories in these notes. They are divided into three parts dealing with spinor helicity, twistors and super-twistors respectively. In the first part, we introduce the off-shell spinor helicity formalism and apply it in several contexts including double copy relations, connection to four dimensional scattering amplitudes, correlators in Chern-Simons matter theories and the holography of chiral higher spin theory. The second part of the notes introduces the twistor space formalism. After discussing the geometry of twistor space, we derive the Penrose transform for conserved currents, scalars with arbitrary scaling dimension as well as generic non-conserved operators. We also explicitly show how the spinor and twistor approaches are related. We discuss how correlators of these operators and conserved currents in particular drastically simplify in twistor space unveiling their hidden simplicity. We also extend our construction to super-conformal field theories and develop a manifest super-twistor space formalism and derive the supersymmetric Penrose transform. We find that the supersymmetric correlators are simple and natural generalizations of their non-supersymmetric counterparts. The notes are made to be self-contained and also include over $50$ exercises that illustrate the formalism.

• The paper develops an off-shell spinor helicity and twistor formalism to simplify the computation of 3D CFT correlators.
• It constructs explicit two- and three-point functions with detailed analysis of parity-odd structures and double copy features.
• The work extends to supersymmetric frameworks, offering new insights for analytic bootstrap, holography, and higher-spin dualities.

Spinor and Twistor Formalism in 3D Conformal Field Theory: A Technical Overview

Introduction and Motivation

The lecture notes "Lectures on the Spinor and Twistor Formalism in 3D Conformal Field Theory" provide a comprehensive and technically detailed exposition of the application of spinor helicity, twistor, and super-twistor methods to three-dimensional conformal field theories (CFTs). The work is structured in three main parts: the development of off-shell spinor helicity variables for 3D CFTs, the formulation of twistor and super-twistor frameworks, and the explicit construction of correlators and conformal invariants in these languages. The notes emphasize the unification of techniques from the modern amplitudes program with the paper of CFT correlators, particularly in the context of AdS/CFT and higher-spin holography.

Spinor Helicity Formalism in 3D

Off-Shell Spinor Helicity Variables

The first part of the notes systematically develops the off-shell spinor helicity formalism for 3D QFTs, both in Lorentzian ($\mathbb{R}^{2,1}$) and Euclidean ($\mathbb{R}^3$) signatures. The construction leverages the isomorphisms $SO(2,1) \cong SL(2,\mathbb{R})/\mathbb{Z}_2$ and $SO(3) \cong SU(2)/\mathbb{Z}_2$ to represent 3-momenta as symmetric bispinors. The momentum $p_\mu$ is encoded as $p_{a}^{b} = (\sigma^\mu)^b_a p_\mu$, and decomposed in terms of two real (Lorentzian) or complex conjugate (Euclidean) spinors $\lambda_a$, $\bar{\lambda}_a$:

$$p_a^b = \frac{1}{2}(\lambda_a \bar{\lambda}^b + \lambda^b \bar{\lambda}_a)$$

The formalism naturally incorporates the little group redundancies and reality conditions appropriate to each signature.

Helicity Basis and Correlators

The helicity basis is constructed by contracting conserved currents with polarization spinors derived from $\lambda$ and $\bar{\lambda}$. This reduces the tensorial structure of spin-$s$ currents to two independent components, $J_s^\pm$, corresponding to helicities $\pm s$. The conformal Ward identities are then recast in spinor helicity variables, with explicit forms for the action of translations, rotations, dilatations, and special conformal transformations. The SCT generator acts as a second-order differential operator, and its nontrivial action encodes the Ward-Takahashi identities for current conservation.

Structure of Two- and Three-Point Functions

The general forms of two- and three-point functions of conserved currents are derived, with explicit dependence on spinor brackets and momentum invariants. Notably, in 3D, parity-odd structures appear already at the level of two-point functions, a feature absent in higher dimensions. The three-point functions are classified into homogeneous (identically conserved) and non-homogeneous (saturating the Ward identity) solutions, with the homogeneous sector corresponding to higher-derivative bulk interactions in AdS$_4$ and the non-homogeneous sector to minimal couplings.

The formalism also elucidates the double copy structure of three-point correlators, the flat space limit (recovering 4D S-matrix elements), and the mapping between free bosonic/fermionic correlators and the general CFT solution.

Twistor and Dual Twistor Formalism

Twistor Geometry and Incidence Relations

The second part introduces the real twistor space $\mathbb{PT} \subset \mathbb{RP}^3$, with projective coordinates $Z^A = (\lambda^a, \bar{\mu}_{a'})$ in the fundamental of $Sp(4)$, the double cover of the 3D conformal group $SO(3,2)$. The incidence relation $\bar{\mu}_a = -x_{ab} \lambda^b$ encodes the correspondence between points in spacetime and lines in twistor space, and vice versa, with null separation arising naturally from the intersection of lines.

Penrose and Witten Transforms

The Penrose transform is formulated for symmetric traceless conserved currents:

$$J_s^{a_1 \cdots a_{2s}}(x) = \int \langle \lambda d\lambda \rangle \lambda^{a_1} \cdots \lambda^{a_{2s}} \hat{J}_s^+(\lambda, \bar{\mu})|_X$$

with projective homogeneity fixed by conformal invariance. The Witten half-Fourier transform connects the spinor helicity and twistor representations, and the equivalence of the Penrose and Witten approaches is established via explicit integration and projective identities.

Conformal Invariants and Ward Identities

The conformal generators act as first-order differential operators on twistor space, and the conformal Ward identities are solved by functions of symplectic dot products and projective delta functions. The explicit construction of two- and three-point functions in twistor space reveals a drastic simplification: the parity-even sector is captured by rational functions of twistor invariants, while the parity-odd sector requires the introduction of the infinity twistor $I_{AB}$, breaking conformal invariance down to the Poincaré subgroup and encoding contact terms and epsilon structures.

Generalization to Scalars and Generic Operators

The Penrose transform is extended to scalar operators of arbitrary scaling dimension, with the action of the conformal generators involving nonlocal (inverse derivative) terms. The two-point function of scalars in twistor space necessarily involves the infinity twistor, reflecting the need for dimensionful invariants.

Super-Twistor Formalism and Supersymmetric Extensions

Super-Twistor Geometry and Super-Penrose Transform

The super-twistor space $\mathbb{RP}^{3|\mathcal{N}}$ is constructed for $\mathcal{N}$-extended supersymmetry, with coordinates $(Z^A, \psi^N)$ in the fundamental of $OSp(\mathcal{N}|4;\mathbb{R})$. The super-Penrose transform is derived for conserved supercurrents, with the super-incidence relations encoding both spacetime and fermionic coordinates. The super-Witten transform is constructed in parallel, and the equivalence with the super-Penrose approach is established.

Superconformal Ward Identities and Correlators

The $OSp(\mathcal{N}|4)$ generators act as graded differential operators on super-twistor space, and the superconformal Ward identities are solved by functions of super-symplectic invariants and super-projective delta functions. The two- and three-point functions of supercurrents are shown to be natural supersymmetric extensions of their non-supersymmetric counterparts, with the Grassmann structure enforcing selection rules on the allowed correlators.

Parity-Odd Supercorrelators and the Super-Infinity Twistor

The parity-odd sector in the supersymmetric context is constructed via a super-epsilon transform, with the super-infinity twistor entering the structure of contact terms and parity-odd invariants. The explicit form of two- and three-point supercorrelators is given, and the extension to super-scalars and generic superprimaries is outlined.

Implications and Future Directions

Theoretical and Practical Implications

The formalism developed in these notes provides a unified and computationally efficient framework for the paper of 3D CFT correlators, with direct applications to AdS$_4$/CFT$_3$ holography, higher-spin theories, and the conformal bootstrap. The spinor helicity and twistor approaches drastically simplify the tensorial complexity of spinning correlators, making manifest the underlying conformal and higher-spin symmetries. The explicit connection to bulk AdS amplitudes, double copy structures, and the flat space limit enables a direct translation between CFT data and S-matrix elements.

The extension to super-twistor space opens the door to a systematic analysis of superconformal correlators, with potential applications to the classification of superconformal field theories, the paper of protected sectors, and the construction of manifestly supersymmetric bootstrap equations.

Open Problems and Future Developments

Several open directions are highlighted by the work:

• Higher-Point and Generic Operator Correlators: While the formalism for two- and three-point functions is complete, the explicit construction of four- and higher-point spinning correlators in twistor space, especially for generic operators, remains an open problem. The nonlocal action of the conformal generators in these cases presents technical challenges.
• Conformal Bootstrap in Twistor Space: The manifestly conformal and projective nature of the twistor formalism suggests a natural stage for the spinning conformal bootstrap, potentially enabling new analytic and numerical approaches to the classification of 3D CFTs.
• Holography and Higher-Spin Dualities: The identification of chiral and anti-chiral subsectors in Chern-Simons matter theories as duals to chiral higher-spin gravity in AdS$_4$ provides a concrete realization of non-unitary holography. The extension of these results to loop corrections, nontrivial topologies, and nonperturbative effects is a promising direction.
• Supersymmetric Generalizations: The extension to higher $\mathcal{N}$, the classification of superconformal invariants, and the explicit construction of superconformal blocks in twistor space are natural next steps.

Conclusion

The lecture notes provide a technically robust and conceptually unified treatment of spinor and twistor methods in 3D conformal field theory, with explicit constructions of correlators, conformal invariants, and Ward identities in both non-supersymmetric and supersymmetric settings. The formalism not only streamlines computations but also reveals deep structural connections between CFT correlators, scattering amplitudes, and holographic dualities. The approach is poised to play a central role in future developments in the analytic conformal bootstrap, higher-spin holography, and the paper of superconformal field theories in three dimensions.