[Paper Review] Embedding vs. 6D twistors

Warren Siegel|arXiv (Cornell University)|Apr 25, 2012

Algebraic structures and combinatorial models9 references11 citations

TL;DR

This paper compares the embedding formalism and 6D twistor methods for realizing conformal symmetry in 4D field theories, showing that 6D twistors provide a more natural framework for handling spin and supersymmetry by encoding conformal invariance through spinor variables and solving algebraic constraints via 6D twistors. The key contribution is a manifestly conformal formulation of field operators using 4D spinor indices via 6D supertwistors, with explicit constructions for N=3 super Yang-Mills and generalizations to chiral, chiral analytic, and real analytic superspaces.

ABSTRACT

We review the relation between the "embedding" formalism and spinorial projective space. The latter is more convenient when treating spin (and indispensable for supersymmetry), as it maintains manifest conformal symmetry while using 4-dimensional indices on fields/operators. It does this by solving all algebraic constraints using 6-dimensional (off-shell) twistors. In an added note we review the supersymmetric generalization, and give some new results for N=3.

Motivation & Objective

• To clarify the relationship between the embedding formalism and spinorial projective space in conformal field theory.
• To demonstrate that 6D twistors provide a more efficient and manifestly conformal framework for handling spin and supersymmetry than the embedding formalism.
• To generalize the twistor construction to supersymmetric theories, particularly for N=3, and derive new results in chiral and analytic superspaces.
• To show that field operators with 4D spinor indices can be consistently described using 6D supertwistors without solving algebraic constraints explicitly.
• To establish a systematic construction of conformal invariants and correlators using superdeterminants of twistor matrices in various supersymmetric settings.

Proposed method

• Uses 6D (super)twistors as rectangular matrices λ and ¯λ with global SU(2,2|N) and local GL(2|n,C) indices to encode spacetime and spinor structure.
• Solves the null constraint λ¯λ = 0 via a gauge-fixed parametrization involving compensator matrices u and ¯u, and matrices w, ˜w, ˜¯w for different supersymmetric sectors.
• Constructs field operators Φ as homogeneous functions of λ and ¯λ with definite symmetry and homogeneity, reducing to 4D spinor-tensor fields via the parametrization.
• Derives conformal invariants from the superdeterminant of λ′¯λ, which encodes free propagators and multi-point correlators with correct scale weights.
• Distinguishes three cases: chiral (n=0), chiral analytic (n=(N−1)/2, N odd), and real analytic (n=N/2, N even), each with distinct index structures and operator dependencies.
• Applies the formalism to N=3 super Yang-Mills, showing the field strength lives on chiral analytic space while prepotentials live on the full superspace.

Experimental results

Research questions

• RQ1 How does the 6D twistor formalism compare to the embedding formalism in preserving manifest conformal symmetry while handling spin degrees of freedom in 4D field theories?
• RQ2 What is the role of 6D twistors in simplifying the treatment of spin and supersymmetry compared to conventional 4D spinor or vector formulations?
• RQ3 How are conformal invariants and correlators constructed in the 6D twistor framework, and what is their relation to free propagators and scale weights?
• RQ4 What are the correct index structures and parametrizations for chiral, chiral analytic, and real analytic superspaces in the N=3 case?
• RQ5 How does the local GL(2|n,C) gauge symmetry constrain the form of operators and correlators in different supersymmetric sectors?

Key findings

• The 6D twistor formalism provides a manifestly conformal description of 4D field operators using 4D spinor indices, avoiding the need to solve algebraic constraints explicitly.
• Field operators are constructed as homogeneous functions of λ and ¯λ with definite symmetry, reducing to 4D spinor-tensor fields via the parametrization λ = u(I −x), ¯λ = (x I)¯u.
• For N=3, the field strength of super Yang-Mills lives on the chiral analytic superspace, while prepotentials live on the larger full superspace, consistent with known results.
• Conformal invariants are built from the superdeterminant of λ′¯λ, which gives the correct free propagators for scalar field strengths in N=0,1,2,3,4 superspaces.
• In chiral and chiral analytic cases, operators carry SL(2,C) or SL(2|n,C) indices respectively, and the 2-point function is given by ˆx(µ.σ...ˆxν).τ / (ˆx²)˜δ with ˆx = x − x′ + θ¯θ′.
• For N≠0,4, a U(1) selection rule restricts correlators in the full (chiral + antichiral) space, reflecting the structure of R-symmetry and gauge invariance.