${\mathbb{Z}}_2$-orbifold construction associated with $(-1)$-isometry and uniqueness of holomorphic vertex operator algebras of central charge 24

Abstract

The vertex operator algebra structure of a strongly regular holomorphic vertex operator algebra $V$ of central charge $24$ is proved to be uniquely determined by the Lie algebra structure of its weight one space $V_1$ if $V_1$ is a Lie algebra of the type $A_{1,4}^{12}$, $B_{2,2}^6$, $B_{3,2}^4$, $B_{4,2}^3$, $B_{6,2}^2$, $B_{12,2}$, $D_{4,2}^2{B}_{2,1}^4$, $D_{8,2}{B}_{4,1}^2$, $A_{3,2}^4{A}_{1,1}^4$, $D_{5,2}^2{A}_{3,1}^2$, $D_{9,2}{A}_{7,1}$, $C_{4,1}^4$ or $D_{6,2}{B}_{3,1}^2{C}_{4,1}$.

15 pages (minor revision. organization of the whole paper is changed)