Generalization of the Knizhnik-Zamolodchikov-Equations

TL;DR

This paper generalizes the Knizhnik-Zamolodchikov equations from affine Lie algebras to broader conformal field theories, enabling analysis of correlation functions beyond rational models.

Contribution

It introduces a new framework for deriving differential equations for correlation functions in non-rational conformal field theories using Nahm's small spaces.

Findings

Derived first order differential equations for generalized models

Analyzed properties of the associated connections

Illustrated with examples from Virasoro minimal models

Abstract

In this letter we introduce a generalization of the Knizhnik- Zamolodchikov equations from affine Lie algebras to a wide class of conformal field theories (not necessarily rational). The new equations describe correlations functions of primary fields and of a finite number of their descendents. Our proposal is based on Nahm's concept of small spaces which provide adequate substitutes for the lowest energy subspaces in modules of affine Lie algebras. We explain how to construct the first order differential equations and investigate properties of the associated connections, thereby preparing the grounds for an analysis of quantum symmetries. The general considerations are illustrated in examples of Virasoro minimal models.