Harmonic Analysis and Mean Field Theory

Abstract

We review some aspects of harmonic analysis for the Euclidean conformal group, including conformally-invariant pairings, the Plancherel measure, and the shadow transform. We introduce two efficient methods for computing these quantities: one based on weight-shifting operators, and another based on Fourier space. As an application, we give a general formula for OPE coefficients in Mean Field Theory (MFT) for arbitrary spinning operators. We apply this formula to several examples, including MFT for fermions and "seed" operators in 4d, and MFT for currents and stress-tensors in 3d.

arXiv comment58 pages + appendices

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• A Generalization of Sachdev-Ye-Kitaev
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• The 3d Stress-Tensor Bootstrap
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• General Three-Point Functions in 4D CFT
• Casimir recursion relations for general conformal blocks
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• Conformal Block Expansion in Celestial CFT
• Unitarity Methods in AdS/CFT
• The Gross-Neveu-Yukawa Archipelago
• A Derivation of AdS/CFT for Vector Models
• Harmonic analysis of 2d CFT partition functions
• The Källén-Lehmann representation in de Sitter spacetime
• Celestial Blocks and Transverse Spin in the Three-Point Energy Correlator
• Detectors in weakly-coupled field theories
• Rigorous bounds on irrelevant operators in the 3d Ising model CFT
• Graviton partial waves and causality in higher dimensions
• Lightcone Bootstrap at higher points
• Scattering Equations in AdS: Scalar Correlators in Arbitrary Dimensions
• Universal Asymptotics for High Energy CFT Data
• On loop celestial amplitudes for gauge theory and gravity
• Conformal 3-point functions and the Lorentzian OPE in momentum space
• Recursion relation for general 3d blocks
• Three-point energy correlators and the celestial block expansion
• Bootstrapping $N_f=4$ conformal QED${}_3$
• From bulk loops to boundary large-N expansion
• A note on the representations of $\text{SO}(1,d+1)$
• Recursion relations for 5-point conformal blocks
• Helicity basis for three-dimensional conformal field theory
• Heavy-light Bootstrap from Lorentzian Inversion Formula
• New Methods for Conformal Correlation Functions
• Holographic cameras: an eye for the bulk
• Opacity from Loops in AdS
• Convergent Momentum-Space OPE and Bootstrap Equations in Conformal Field Theory
• On Galilean conformal bootstrap
• A microscopic realization of dS${}_3$
• Spectrum continuity and level repulsion: the Ising CFT from infinitesimal to finite $\mathbf{\epsilon }$
• Holographic solar systems and hydrogen atoms: non-relativistic physics in AdS and its CFT dual
• blocks_3d: Software for general 3d conformal blocks
• Unitarity at the Late time Boundary of de Sitter
• The Shadow Formalism of Galilean CFT${}_2$
• The F-theorem in the melonic limit
• The perturbative CFT optical theorem and high-energy string scattering in AdS at one loop
• Conformal partial waves in momentum space
• Remarks on a melonic field theory with cubic interaction
• Exploring Confinement in Anti-de Sitter Space
• Flat-space structure of gluon and graviton in AdS
• The Inversion Formula and 6j Symbol for 3d Fermions
• Instability of complex CFTs with operators in the principal series
• Finite-size versus finite-temperature effects in the critical long-range $O(N)$ model
• Hidden Relations of Central Charges and OPEs in Holographic CFT
• Anomalous Dimensions from Thermal AdS Partition Functions
• Large $N$ analytical functional bootstrap I: 1D CFTs and total positivity
• A Large Twist Limit for Any Operator
• Spinors and conformal correlators
• Crossing Symmetry for Long Multiplets in 4D $\mathcal{N}=1$ SCFTs
• Conformal bootstrap in momentum space at finite volume
• Probing Stringy Horizons with Pole-Skipping in Non-Maximal Chaotic Systems
• Finite-coupling spectrum of O(N) model in AdS