Non-linear Lie conformal algebras with three generators

TL;DR

This paper classifies certain non-linear Lie conformal algebras with three generators, revealing a new family of algebras and their vertex algebra realizations, including extensions of known affine vertex algebras at critical levels.

Contribution

It introduces a new 1-parameter family of non-linear Lie conformal algebras and constructs their free-field realizations extending Wakimoto's construction.

Findings

Discovery of a 1-parameter family of non-linear Lie conformal algebras R_{-1}^d.

Construction of corresponding vertex algebras V_{-1}^d.

Extension of Wakimoto realization for these algebras.

Abstract

We classify certain non-linear Lie conformal algebras with three generators, which can be viewed as deformations of the current Lie conformal algebra of sl_2. In doing so we discover an interesting 1-parameter family of non-linear Lie conformal algebras R_{-1}^d and the corresponding freely generated vertex algebras V_{-1}^d, which includes for d=1 the affine vertex algebra of sl_2 at the critical level k=-2. We construct free-field realizations of the algebras V_{-1}^d extending the Wakimoto realization of the affine vertex algebra of sl_2 at the critical level, and we compute their Zhu algebras.