Projectors, Shadows, and Conformal Blocks

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.

38 pages, 3 figures; v2 fixed typos, added references; v3 JHEP version; v4, v5, v6 typos and corrections

References in corpus (22)

• Holography from Conformal Field Theory
• Solving the 3D Ising Model with the Conformal Bootstrap
• Conformal collider physics: Energy and charge correlations
• Writing CFT correlation functions as AdS scattering amplitudes
• Bootstrapping the O(N) Vector Models
• Spinning Conformal Blocks
• Bounds on 4D Conformal and Superconformal Field Theories
• Carving Out the Space of 4D CFTs
• A Natural Language for AdS/CFT Correlators
• Radial Coordinates for Conformal Blocks
• Unitarity and the Holographic S-Matrix
• Analyticity and the Holographic S-Matrix
• Bounds in 4D Conformal Field Theories with Global Symmetry
• D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes
• Central Charge Bounds in 4D Conformal Field Theory
• Diagonal Limit for Conformal Blocks in d Dimensions
• Mellin Amplitudes for Dual Conformal Integrals
• Conformal Bootstrap in Three Dimensions?
• Superembedding methods for 4d N=1 SCFTs
• Amplitudes at Weak Coupling as Polytopes in AdS_5
• AdS/CFT in superspace
• The box integrals in momentum-twistor geometry

Cited by in corpus (237)

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• Bootstrapping the O(N) Vector Models
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• Conformal Regge theory
• A Conformal Basis for Flat Space Amplitudes
• EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions
• Bootstrapping Mixed Correlators in the 3D Ising Model
• Bootstrapping Conformal Field Theories with the Extremal Functional Method
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• Radial Coordinates for Conformal Blocks
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• Spinning AdS Propagators
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• AdS Field Theory from Conformal Field Theory
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• $p$-Forms on the Celestial Sphere
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