Weight Shifting Operators and Conformal Blocks

TL;DR

This paper introduces conformally-covariant differential operators and a crossing equation that simplify calculations of conformal blocks with spin, providing new formulas and identities useful for conformal field theory analysis.

Contribution

It presents a new class of differential operators and a crossing equation, enabling simplified derivations of conformal blocks and related identities in CFTs.

Findings

Derived a general formula for conformal blocks with arbitrary representations.

Provided new expressions for seed conformal blocks in 3d and 4d CFTs.

Established identities and recursion relations for conformal blocks and crossing kernels.

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.