Reduction of the Knizhnik-Zamolodchikov Equation - a Way of Producing Virasoro Algebra Singular Vectors

TL;DR

This paper demonstrates how the sl(2,C) Knizhnik-Zamolodchikov equation relates to Virasoro algebra singular vectors, providing a method to convert certain KZ solutions into Virasoro singular vectors, revealing new connections between these algebraic structures.

Contribution

It introduces an algorithm to convert Malikov-Feigin-Fuks singular vectors into Virasoro algebra singular vectors based on KZ equations.

Findings

KZ equations recover Virasoro singular vectors up to gauge transformations.

Infinite KZ matrix system truncates for Kac-Kazhdan spins due to decoupling.

Algorithm for converting KZ singular vectors into Virasoro vectors is proposed.

Abstract

It is shown that the sl(2,C) KZ equation for (half-) integer isospins recovers, up to a gauge transformation, the matrix system for Virasoro algebra singular vectors of Bauer et al. In the case of Kac-Kazhdan spins the general (infinite matrix) KZ system is truncated due to the decoupling of the A^(1)_1 singular vectors. This suggests an algorithm converting Malikov-Feigin-Fuks singular vectors into Virasoro ones.