Gaudin Models and Multipoint Conformal Blocks: General Theory

Abstract

The construction of conformal blocks for the analysis of multipoint correlation functions with $N>4$ local field insertions is an important open problem in higher dimensional conformal field theory. This is the first in a series of papers in which we address this challenge, following and extending our short announcement in [Phys. Rev. Lett. 126, 021602]. According to Dolan and Osborn, conformal blocks can be determined from the set of differential eigenvalue equations that they satisfy. We construct a complete set of commuting differential operators that characterize multipoint conformal blocks for any number $N$ of points in any dimension and for any choice of OPE channel through the relation with Gaudin integrable models we uncovered in [Phys. Rev. Lett. 126, 021602]. For 5-point conformal blocks, there exist five such operators which are worked out smoothly in the dimension $d$.

52 pages, 10 figures; v2: published version, corrected typos in main text and Mathematica notebook

References in corpus (7)

• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT
• Multipoint Bootstrap I: Light-Cone Snowflake OPE and the WL Origin
• Holographic dual of the five-point conformal block
• From Gaudin Integrable Models to $d$-dimensional Multipoint Conformal Blocks
• Limits of Gaudin Systems: Classical and Quantum Cases
• Inozemtsev System as Seiberg-Witten Integrable system

Cited by in corpus (21)

• Selected Topics in Analytic Conformal Bootstrap: A Guided Journey
• Lightcone Bootstrap at higher points
• OPE statistics from higher-point crossing
• Gaudin Models and Multipoint Conformal Blocks II: Comb channel vertices in 3D and 4D
• Recursion relations for 5-point conformal blocks
• Gaudin Models and Multipoint Conformal Blocks III: Comb channel coordinates and OPE factorisation
• Multipoint correlators on the supersymmetric Wilson line defect CFT
• Shadow Celestial Amplitude
• Improving the five-point bootstrap
• Lining up a Positive Semi-Definite Six-Point Bootstrap
• Feynman Rules for Scalar Conformal Blocks
• Universal Spinning Casimir Equations and Their Solutions
• Quantum nonequilibrium dynamics from Knizhnik-Zamolodchikov equations
• Conformal Four-Point Integrals: Recursive Structure, Toda Equations and Double Copy
• Stampedes II: Null Polygons in Conformal Gauge Theory
• Conformal Blocks in Two and Four Dimensions from Oscillator Representations
• BCJ Amplitude Relations for Anti-de Sitter Boundary Correlators in Embedding Space
• Spinors and conformal correlators
• Separation of variables for higher rank integrable models: review of recent progress
• Recursive structure of the Gauß hypergeometric function and boundary/crosscap conformal block
• Hypergeometry from $\hat{\mathrm{P}}$-Symmetry: Feynman Integrals in One and Two Dimensions