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Projectors, Shadows, and Conformal Blocks

David Simmons-Duffin

TL;DR

The paper presents a unified, embedding-space shadow formalism to compute conformal blocks for operators in arbitrary Lorentz representations across dimensions, enabling bootstrap analyses beyond scalars. It develops conformal integrals, Casimir eigenvalue structure, and monodromy projections, along with explicit tensor and spinor (twistor) formalisms in 4d, demonstrated by concrete examples such as antisymmetric tensor exchange. The approach yields algorithmic, potentially automatable expressions for higher-spin blocks and links naturally to Mellin-space formulations, with avenues toward superconformal extensions. Collectively, this framework broadens the reach of the conformal bootstrap to spinning operators and provides practical tools for systematic bootstrap constraints.

Abstract

We introduce a method for computing conformal blocks of operators in arbitrary Lorentz representations in any spacetime dimension, making it possible to apply bootstrap techniques to operators with spin. The key idea is to implement the "shadow formalism" of Ferrara, Gatto, Grillo, and Parisi in a setting where conformal invariance is manifest. Conformal blocks in $d$-dimensions can be expressed as integrals over the projective null-cone in the "embedding space" $\mathbb{R}^{d+1,1}$. Taking care with their analytic structure, these integrals can be evaluated in great generality, reducing the computation of conformal blocks to a bookkeeping exercise. To facilitate calculations in four-dimensional CFTs, we introduce techniques for writing down conformally-invariant correlators using auxiliary twistor variables, and demonstrate their use in some simple examples.

Projectors, Shadows, and Conformal Blocks

TL;DR

The paper presents a unified, embedding-space shadow formalism to compute conformal blocks for operators in arbitrary Lorentz representations across dimensions, enabling bootstrap analyses beyond scalars. It develops conformal integrals, Casimir eigenvalue structure, and monodromy projections, along with explicit tensor and spinor (twistor) formalisms in 4d, demonstrated by concrete examples such as antisymmetric tensor exchange. The approach yields algorithmic, potentially automatable expressions for higher-spin blocks and links naturally to Mellin-space formulations, with avenues toward superconformal extensions. Collectively, this framework broadens the reach of the conformal bootstrap to spinning operators and provides practical tools for systematic bootstrap constraints.

Abstract

We introduce a method for computing conformal blocks of operators in arbitrary Lorentz representations in any spacetime dimension, making it possible to apply bootstrap techniques to operators with spin. The key idea is to implement the "shadow formalism" of Ferrara, Gatto, Grillo, and Parisi in a setting where conformal invariance is manifest. Conformal blocks in -dimensions can be expressed as integrals over the projective null-cone in the "embedding space" . Taking care with their analytic structure, these integrals can be evaluated in great generality, reducing the computation of conformal blocks to a bookkeeping exercise. To facilitate calculations in four-dimensional CFTs, we introduce techniques for writing down conformally-invariant correlators using auxiliary twistor variables, and demonstrate their use in some simple examples.

Paper Structure

This paper contains 25 sections, 131 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Constructing Conformal Blocks
  3. Defining Properties
  4. The Embedding Space
  5. Conformal Integrals
  6. The Conformal Casimir
  7. Consistency with the OPE
  8. Projectors and Shadows
  9. Conformal Integrals and Monodromy Invariants
  10. Scalar Four-point Integrals
  11. Tensor Four-point Integrals
  12. Higher Spin Conformal Blocks
  13. General Method
  14. Tensor Operators in the Embedding Space
  15. Projectors for Tensor Operators
  16. ...and 10 more sections

Figures (3)

  • Figure 1: We rotate the contour for the integral (\ref{['eq:monodromyready']}) in the complex $\gamma$-plane so that it passes along the negative real axis. It breaks up into $I_1,I_2,I_3$ as shown.
  • Figure 2: This contour is contractible on the punctured Riemann sphere, and thus integrates to zero. However, it can also be written as a linear combination of the segments $I_1,I_2,I_3$ as shown. The associated phases are determined by the monodromy around the branch points $0,\gamma_1,\gamma_2,\infty$.
  • Figure 3: The monodromy $M$ moves $\gamma_1$ twice around the origin, so that $I_1$ picks up a phase $e^{2i\phi_0}$, while $I_3$ remains invariant.