Gaudin Models and Multipoint Conformal Blocks: General Theory

TL;DR

This paper develops a comprehensive mathematical framework for constructing multipoint conformal blocks in higher-dimensional conformal field theories, extending previous work by linking them to Gaudin integrable models.

Contribution

It introduces a complete set of commuting differential operators for multipoint conformal blocks, generalizing to any number of points, dimensions, and OPE channels, based on Gaudin models.

Findings

Constructed differential operators for multipoint conformal blocks.

Extended the framework to any number of points and dimensions.

Explicitly worked out 5-point conformal blocks in any dimension.

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 $d$.