Weight Shifting Operators and Conformal Blocks

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.

84 pages; v2: fix 6j example in 3d

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• Spinning Conformal Correlators
• Analyticity in Spin in Conformal Theories
• Projectors, Shadows, and Conformal Blocks
• The Bootstrap Program for Boundary CFT_d
• Conformal Regge theory
• A Conformal Basis for Flat Space Amplitudes
• Spinning Conformal Blocks
• Bootstrapping Mixed Correlators in the 3D Ising Model
• OPE Convergence in Conformal Field Theory
• Bounds on 4D Conformal and Superconformal Field Theories
• Carving Out the Space of 4D CFTs
• Bootstrapping Conformal Field Theories with the Extremal Functional Method
• Radial Coordinates for Conformal Blocks
• Large Spin Perturbation Theory
• Averaged Null Energy Condition from Causality
• Causality Constraints in Conformal Field Theory
• Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks
• The $(2,0)$ superconformal bootstrap
• A Stereoscopic Look into the Bulk
• Six-dimensional Methods for Four-dimensional Conformal Field Theories
• Universal Constraints on Conformal Operator Dimensions
• More constraining conformal bootstrap
• Bootstrapping the 3d Ising twist defect
• More on Supersymmetric and 2d Analogs of the SYK Model
• A Mellin space approach to the conformal bootstrap
• Bootstrapping the O(N) Archipelago
• Bounds in 4D Conformal Field Theories with Global Symmetry
• The $\mathcal{N}=8$ Superconformal Bootstrap in Three Dimensions
• A Proof of the Conformal Collider Bounds
• The $\mathcal{N}=2$ superconformal bootstrap
• Bootstrapping 3D Fermions
• Central Charge Bounds in 4D Conformal Field Theory
• Rigorous Limits on the Interaction Strength in Quantum Field Theory
• Einstein gravity 3-point functions from conformal field theory
• Recursion Relations for Conformal Blocks
• Improved bounds for CFT's with global symmetries
• Deep Inelastic Scattering in Conformal QCD
• Bootstrapping phase transitions in QCD and frustrated spin systems
• Five dimensional $O(N)$-symmetric CFTs from conformal bootstrap
• Solving CFTs with Weakly Broken Higher Spin Symmetry
• Bootstrapping $O(N)$ Vector Models in $4<d<6$
• Superintegrability of $d$-dimensional Conformal Blocks
• Conformal Bootstrap With Slightly Broken Higher Spin Symmetry
• Conformal correlators of mixed-symmetry tensors
• Spinning Witten Diagrams
• Crossing Symmetry in Alpha Space
• Approaching conformal window of $O(n)\times O(m)$ symmetric Landau-Ginzburg models from conformal bootstrap
• Projectors and seed conformal blocks for traceless mixed-symmetry tensors
• Conformal Collider Physics from the Lightcone Bootstrap
• Conformal Bootstrap in the Regge Limit
• More $\mathcal{N}=4$ superconformal bootstrap
• The superconformal bootstrap for structure constants
• Towards Bootstrapping QED${}_3$
• Radial expansion for spinning conformal blocks
• Carving Out the End of the World or (Superconformal Bootstrap in Six Dimensions)
• Seed Conformal Blocks in 4D CFT
• PyCFTBoot: A flexible interface for the conformal bootstrap
• Harmony of Spinning Conformal Blocks
• Bootstrapping Mixed Correlators in the Five Dimensional Critical O(N) Models
• General Three-Point Functions in 4D CFT
• A New Spin on Causality Constraints
• The analytic structure of conformal blocks and the generalized Wilson-Fisher fixed points
• Scalar-Vector Bootstrap
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• Bounds on OPE Coefficients in 4D Conformal Field Theories
• Geodesic Diagrams, Gravitational Interactions & OPE Structures
• Long Multiplet Bootstrap
• More on analytic bootstrap for O(N) models
• Cut-touching linear functionals in the conformal bootstrap
• Mass Ladder Operators from Spacetime Conformal Symmetry
• Spinning Geodesic Witten Diagrams
• Exploring the Minimal 4D $\mathcal{N}=1$ SCFT
• Comments on Determinant Formulas for General CFTs
• Crossing symmetry and Higher spin towers
• Quantum Integrable Systems from Conformal Blocks
• SO_0(1,d+1) Racah coefficients: Type I representations
• Determinant Formula for Parabolic Verma Modules of Lie Superalgebras

Cited by in corpus (90)

• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Light-ray operators in conformal field theory
• A spacetime derivation of the Lorentzian OPE inversion formula
• An introduction to the SYK model
• $d$-dimensional SYK, AdS Loops, and $6j$ Symbols
• From dS to AdS and back
• Linking the Singularities of Cosmological Correlators
• Dispersive CFT Sum Rules
• Harmonic Analysis and Mean Field Theory
• Multipoint Conformal Blocks in the Comb Channel
• Selected Topics in Analytic Conformal Bootstrap: A Guided Journey
• Shocks, Superconvergence, and a Stringy Equivalence Principle
• Boostless Cosmological Collider Bootstrap
• Quantum Interference in Jet Substructure from Spinning Gluons
• Universality at large transverse spin in defect CFT
• Beyond $a=c$: Gravitational Couplings to Matter and the Stress Tensor OPE
• Unitarity Methods in AdS/CFT
• Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications
• Spheres, Charges, Instantons, and Bootstrap: A Five-Dimensional Odyssey
• Recursion Relations in Witten Diagrams and Conformal Partial Waves
• Spinning operators and defects in conformal field theory
• Exact Correlation Functions in Conformal Fishnet Theory
• From Spinning Conformal Blocks to Matrix Calogero-Sutherland Models
• Veneziano Amplitude of Vasiliev Theory
• Spinning Gluons from the QCD Light-Ray OPE
• Integrability of Conformal Blocks I: Calogero-Sutherland Scattering Theory
• Amplitudes meet Cosmology: A (Scalar) Primer
• The graviton four-point function in de Sitter space
• Celestial Blocks and Transverse Spin in the Three-Point Energy Correlator
• Graviton partial waves and causality in higher dimensions
• Double copy structure of parity-violating CFT correlators
• Lightcone Bootstrap at higher points
• Scalar CFTs and Their Large N Limits
• Gaudin Models and Multipoint Conformal Blocks: General Theory
• The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization
• AdS Weight Shifting Operators
• Conformal 3-point functions and the Lorentzian OPE in momentum space
• Recursion relation for general 3d blocks
• Higher-Point Conformal Blocks in the Comb Channel
• Momentum space parity-odd CFT 3-point functions
• Three-point energy correlators and the celestial block expansion
• Spinning conformal defects
• Gaudin Models and Multipoint Conformal Blocks II: Comb channel vertices in 3D and 4D
• A multipoint conformal block chain in $d$ dimensions
• Recursion relations for 5-point conformal blocks
• Helicity basis for three-dimensional conformal field theory
• Constraining momentum space correlators using slightly broken higher spin symmetry
• Seven-Point Conformal Blocks in the Extended Snowflake Channel and Beyond
• Fermions in Boundary Conformal Field Theory : Crossing Symmetry and $\epsilon$-Expansion
• Six-Point Conformal Blocks in the Snowflake Channel
• Gaudin Models and Multipoint Conformal Blocks III: Comb channel coordinates and OPE factorisation
• A handbook of holographic 4-point functions
• New Methods for Conformal Correlation Functions
• Spinor-Helicity Formalism for Massless Fields in AdS${}_4$ II: Potentials
• Tensors and Spinors in de Sitter Space
• Model-Dependence of Minimal-Twist OPEs in $d>2$ Holographic CFTs
• Towards Feynman rules for conformal blocks
• Mellin Amplitudes for Fermionic Conformal Correlators
• Scalar Blocks as Gravitational Wilson Networks
• More on holographic correlators: Twisted and dimensionally reduced structures
• blocks_3d: Software for general 3d conformal blocks
• Momentum space CFT correlators for Hamiltonian truncation
• The Shadow Formalism of Galilean CFT${}_2$
• AdS/CFT Unitarity at Higher Loops: High-Energy String Scattering
• Soft limits of the wavefunction in exceptional scalar theories
• Efficient Rules for All Conformal Blocks
• The perturbative CFT optical theorem and high-energy string scattering in AdS at one loop
• Conformal Group Theory of Tensor Structures
• Differential operators for superconformal correlation functions
• Dimensional reduction of higher-point conformal blocks
• On Conformal Block, Crossing Kernel and Multi-variable Hypergeometric Functions
• Lightcone expansions of conformal blocks in closed form
• Defect Conformal Blocks from Appell Functions
• Conformal Two-Point Correlation Functions from the Operator Product Expansion
• Flat-space structure of gluon and graviton in AdS
• A Sum Rule for Boundary Contributions to the Trace Anomaly
• More Analytic Bootstrap: Nonperturbative Effects and Fermions
• Bootstrapping Witten diagrams via differential representation in Mellin space
• Critical loop models are exactly solvable
• The Inversion Formula and 6j Symbol for 3d Fermions
• Universal Spinning Casimir Equations and Their Solutions
• Shift operators from the simplex representation in momentum-space CFT
• 3d conformal fields with manifest $sl(2,\mathbb{C})$
• Further Results on a Function Relevant for Conformal Blocks
• Crossing Symmetry for Long Multiplets in 4D $\mathcal{N}=1$ SCFTs
• Harmonic Analysis in d-dimensional Superconformal Field Theory
• Spinors and conformal correlators
• Recursive structure of the Gauß hypergeometric function and boundary/crosscap conformal block
• On the Stress Tensor Light-ray Operator Algebra
• On-shell Correlators and Color-Kinematics Duality in Curved Symmetric Spacetimes