Weight shifting operators and conformal blocks

By D. Karateev, Petr Kravchuk, David Simmons-Duffin. Journal of High Energy Physics, 2017

140 citations recorded by the source corpus.

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 deriving 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.

Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/5d8d4a79ec2bdeffe101e981a22d2872dbafa492

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Citations in the corpus

The cosmological bootstrap: inflationary correlators from symmetries and singularities (2018, unknown): “reover, an operator that shifts the weight of the internal eld was introduced in (4.58). Similar weight-shifting operators for conformal correlators in position space have recently been discovered in [91,92]. 45 As we prove in the insert below, the two currents are related as follows J˚ i ˇU 1;1 12 ”
The Cosmological Bootstrap: Spinning correlators from symmetries and factorization (2020, background): “[57, 58]).”
The cosmological bootstrap: weight-shifting operators and scalar seeds (2019, methodology): “In this paper, we use a group-theoretic approach inspired by known tools of conformal field theory [62, 63, 65–67] to provide a new and more elegant derivation of the relevant spin-raising operatorsD uv and polarization sums ΠS,m (see Section 4).”
Light-ray operators in conformal field theory (2018, methodology): “We also note that the crossing equation for weight-shifting operators acting on a twopoint function [57] can be written in terms of shadow transform and conjugation.”
The conformal bootstrap: Theory, numerical techniques, and applications (2018, background): “Karateev et al. (2018) introduced a more general class of “weight-shifting” operators which act on correlation functions.”