On the Constructions of Holomorphic Vertex Operator Algebras of Central Charge 24

In this article, we construct explicitly several holomorphic vertex operator algebras of central charge 24 using Virasoro frames. The Lie algebras associated to their weight one subspaces are of the types A_1,2 A_3,4⁴, A_1,2D_5,8, A_1,1³A_7,4{A_{1,2} {A_{3,4}}^4, A_{1,2}D_{5,8}, {A_{1,1}}^3A_{7,4}} , A_1,1² C_3,2 D_5,4, A_2,1² A_5,2² C_2,1, A_3,1 A_7,2 C_3,1², A_3,1C_7,2{{A_{1,1}}^2 C_{3,2} D_{5,4}, {A_{2,1}}^2 {A_{5,2}}^2 C_{2,1}, A_{3,1} A_{7,2} {C_{3,1}}^2, A_{3,1}C_{7,2}} , A_4,1 A_9,2B_3,1, B_4,1 C_6,1²{A_{4,1} A_{9,2}B_{3,1}, B_{4,1} {C_{6,1}}^2} and B _6,1 C _10,1. These vertex operator algebras correspond to number 7, 10, 18, 19, 26, 33, 35, 40, 48 and 56 in Schellekens’ list Schellekens (Commun Math Phys 153:159–185, 1993).