On rank-two parabolic-type vertex operator algebras

Abstract: In this paper, we undertake a systematic study of the parabolic-type sub-vertex operator algebras (VOAs) (V_P) of rank-two lattice VOAs (V_L), originally introduced by the first-named author. We first classify all possible types of such subVOAs by analyzing the corresponding submonoids (P \subseteq L). For each type of (V_P), we then classify the irreducible modules and determine the fusion rules among them. Finally, we show that the simple quotient (V_H) of any parabolic-type subVOA (V_P) is a (C_1)-cofinite, irrational VOA satisfying the strongly unital property recently introduced by Damiolini--Gibney--Krashen.