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Weight Shifting Operators and Conformal Blocks

Denis Karateev, Petr Kravchuk, David Simmons-Duffin

TL;DR

<3-5 sentence high-level summary>The paper develops a comprehensive framework of weight-shifting operators, built from tensoring with finite-dimensional conformal representations, and a crossing mechanism expressed via 6j symbols. This yields a powerful method to compute conformal blocks with spin by reducing them to derivatives of scalar blocks and to construct seed blocks in 3d and 4d, with explicit recursion relations. The approach unifies embedding-space, diagrammatic, and Verma-module perspectives, enabling spinning-down, inversion formulas, and potential extensions to superconformal theories. Together, these tools broaden analytic control over spinning conformal blocks and open avenues for both analytical and numerical bootstrap applications.

Abstract

We introduce a large class of conformally-covariant differential operators and a crossing equation that they obey. Together, these tools dramatically simplify calculations involving operators with spin in conformal field theories. As an application, we derive a formula for a general conformal block (with arbitrary internal and external representations) in terms of derivatives of blocks for external scalars. In particular, our formula gives new expressions for "seed conformal blocks" in 3d and 4d CFTs. We also find simple derivations of identities between external-scalar blocks with different dimensions and internal spins. We comment on additional applications, including derivation of recursion relations for general conformal blocks, reducing inversion formulae for spinning operators to inversion formulae for scalars, and deriving identities between general 6j symbols (Racah-Wigner coefficients/"crossing kernels") of the conformal group.

Weight Shifting Operators and Conformal Blocks

TL;DR

<3-5 sentence high-level summary>The paper develops a comprehensive framework of weight-shifting operators, built from tensoring with finite-dimensional conformal representations, and a crossing mechanism expressed via 6j symbols. This yields a powerful method to compute conformal blocks with spin by reducing them to derivatives of scalar blocks and to construct seed blocks in 3d and 4d, with explicit recursion relations. The approach unifies embedding-space, diagrammatic, and Verma-module perspectives, enabling spinning-down, inversion formulas, and potential extensions to superconformal theories. Together, these tools broaden analytic control over spinning conformal blocks and open avenues for both analytical and numerical bootstrap applications.

Abstract

We introduce a large class of conformally-covariant differential operators and a crossing equation that they obey. Together, these tools dramatically simplify calculations involving operators with spin in conformal field theories. As an application, we derive a formula for a general conformal block (with arbitrary internal and external representations) in terms of derivatives of blocks for external scalars. In particular, our formula gives new expressions for "seed conformal blocks" in 3d and 4d CFTs. We also find simple derivations of identities between external-scalar blocks with different dimensions and internal spins. We comment on additional applications, including derivation of recursion relations for general conformal blocks, reducing inversion formulae for spinning operators to inversion formulae for scalars, and deriving identities between general 6j symbols (Racah-Wigner coefficients/"crossing kernels") of the conformal group.

Paper Structure

This paper contains 68 sections, 3 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Weight-shifting operators
  3. Finite-dimensional conformal representations
  4. Tensor products with finite-dimensional representations
  5. Covariant differential operators from tensor products
  6. Algebra of weight-shifting operators
  7. Weight-shifting operators in the embedding space
  8. Weight-shifting operators in the embedding space
  9. General dimensions
  10. 1 dimension
  11. 3 dimensions
  12. 4 dimensions
  13. Crossing for differential operators
  14. Conformally-covariant tensor structures
  15. Tensor structures and diagrams
  16. ...and 53 more sections

Key Result

Theorem 2.1

The conformally-covariant operators $\mathcal{D}^A:[\Delta,\rho]\to [\Delta-i,\lambda]$ associated with $W$ are (for generic $\Delta$) in one-to-one correspondence with the irreducible components in the tensor product decomposition W^*⊗ V_Δ,ρ=_i=-j^j _λ∈ (W_i)^*⊗ρ V_Δ-i,λ.

Theorems & Definitions (5)

  • Theorem 2.1
  • Theorem B.1
  • proof
  • Lemma B.2
  • proof : Proof of lemma \ref{['lem:tautology']}