Solutions of the Knizhnik - Zamolodchikov Equation with Rational Isospins and the Reduction to the Minimal Models

TL;DR

This paper establishes a connection between WZNW conformal theory correlators and Virasoro minimal models by solving the Knizhnik-Zamolodchikov equations with rational levels and isospins, using twisted cohomology techniques.

Contribution

It introduces a method to relate WZNW correlators to minimal models through solutions of KZ equations with rational parameters, extending previous results to broader levels and isospin values.

Findings

Derived relations between WZNW and minimal model correlators.

Extended solutions to arbitrary level k+2 and specific isospin values.

Utilized twisted cohomology to solve KZ equations with rational parameters.

Abstract

In the spirit of the quantum Hamiltonian reduction we establish a relation between the chiral $n$-point functions, as well as the equations governing them, of the $A_1^{(1)}$ WZNW conformal theory and the corresponding Virasoro minimal models. The WZNW correlators are described as solutions of the Knizhnik - Zamolodchikov equations with rational levels and isospins. The technical tool exploited are certain relations in twisted cohomology. The results extend to arbitrary level $k+2 \neq 0$ and isospin values of the type $J=j-j'(k+2)$, $ \ 2j, 2j' \in Z\!\!\!Z_+$.