Gaudin Models and Multipoint Conformal Blocks II: Comb channel vertices in 3D and 4D

Abstract

It was recently shown that multi-point conformal blocks in higher dimensional conformal field theory can be considered as joint eigenfunctions for a system of commuting differential operators. The latter arise as Hamiltonians of a Gaudin integrable system. In this work we address the reduced fourth order differential operators that measure the choice of 3-point tensor structures for all vertices of 3- and 4-dimensional comb channel conformal blocks. These vertices come associated with a single cross ratio. Remarkably, we identify the vertex operators as Hamiltonians of a crystallographic elliptic Calogero-Moser-Sutherland model that was discovered originally by Etingof, Felder, Ma and Veselov. Our construction is based on a further development of the embedding space formalism for mixed-symmetry tensor fields. The results thereby also apply to comb channel vertices of 5- and 6-point functions in arbitrary dimension.

arXiv comment67 pages, 4 figures, v2: published version, inaccuracies corrected, figure and tables added, two Mathematica notebooks added

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• Selected Topics in Analytic Conformal Bootstrap: A Guided Journey
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• Gaudin Models and Multipoint Conformal Blocks III: Comb channel coordinates and OPE factorisation
• Shadow Celestial Amplitude
• Lining up a Positive Semi-Definite Six-Point Bootstrap
• Improving the five-point bootstrap
• Feynman Rules for Scalar Conformal Blocks
• Universal Spinning Casimir Equations and Their Solutions
• Quantum nonequilibrium dynamics from Knizhnik-Zamolodchikov equations
• Stampedes II: Null Polygons in Conformal Gauge Theory
• Conformal Blocks in Two and Four Dimensions from Oscillator Representations
• Recursive structure of the Gauß hypergeometric function and boundary/crosscap conformal block
• Complex crystallographic reflection groups and Seiberg-Witten integrable systems: rank 1 case
• Hypergeometry from $\hat{\mathrm{P}}$-Symmetry: Feynman Integrals in One and Two Dimensions