Conformal subalgebras of lattice vertex algebras

Abstract

In this paper we classify, under certain restrictions, all homogeneous conformal subalgebras $\goth L$ of a lattice vertex superalgebra $V_\Lambda$ corresponding to an integer lattice $\Lambda$. We require that $\goth L$ is graded by an almost finite root system $\Delta\subset \Lambda$ and that $\goth L$ is stable under the action of the Heisenberg conformal algebra $\goth H\subset V_\Lambda$. We also describe the root systems of these subalgebras. The key ingredient of this classification is an infinite type conformal algebra $\goth K$ obtained by the Tits-Kantor-Koeher construction from a certain Jordan conformal triple system $\goth J$. We realize a central extension $\^{\goth K}$ of $\goth K$ inside the fermionic vertex superalgebra $V_\Z$, thus extending the bozon-fermion correspondence.