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

TL;DR

This paper links conformal blocks in higher-dimensional conformal field theory to integrable systems, specifically identifying vertex operators as Hamiltonians of an elliptic Calogero-Moser-Sutherland model, extending the understanding of tensor structures.

Contribution

It introduces a novel connection between conformal block vertices and elliptic integrable models, expanding the embedding space formalism for tensor fields in CFT.

Findings

Vertex operators identified as Hamiltonians of elliptic Calogero-Moser-Sutherland model

Results extend to 5- and 6-point functions in arbitrary dimensions

Provides a new perspective on tensor structures in conformal blocks

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.