INERTIA GROUPS AND UNIQUENESS OF HOLOMORPHIC VERTEX OPERATOR ALGEBRAS

We continue our program on classiffication of holomorphic vertex operator algebras of central charge 24. In this article, we show that there exists a unique strongly regular holomorphic VOA of central charge 24, up to isomorphism, if its weight one Lie algebra has the type C 4,10 , D 7,3 A 3,1 G 2,1 , A 5,6 C 2,3 A 1,2 , A 3,1 C 7,2 , D 5,4 C 3,2 A A 1 , 1 2 , or E 6,4 C 2,1 A 2,1 . As a consequence, we have verified that the isomorphism class of a strongly regular holomorphic vertex operator algebra of central charge 24 is determined by its weight one Lie algebra structure if the weight one subspace is nonzero.