Systematic orbifold constructions of Schellekens' vertex operator algebras from Niemeier lattices
Gerald Höhn, Sven Möller
Abstract
We present a systematic, rigorous construction of all 70 strongly rational, holomorphic vertex operator algebras V $V$ of central charge 24 with non-zero weight-one space V 1 $V_1$ as cyclic orbifold constructions associated with the 24 Niemeier lattice vertex operator algebras V N $V_N$ and certain 226 short automorphisms in Aut ( V N ) $\operatorname{Aut}(V_N)$ . We show that up to algebraic conjugacy these automorphisms are exactly the generalised deep holes, as introduced in Möller and Scheithauer (Ann. of Math. (to appear)), of the Niemeier lattice vertex operator algebras with the additional property that their orders are equal to those of the corresponding outer automorphisms. Together with the constructions in Höhn (2017) and Möller and Scheithauer (Ann. of Math. (to appear)), this gives three different uniform constructions of these vertex operator algebras, which are related through 11 algebraic conjugacy classes in Co 0 $\operatorname{Co}_0$ . Finally, by considering the inverse orbifold constructions associated with the 226 short automorphisms, we give the first systematic proof of the result that each strongly rational, holomorphic vertex operator algebra V $V$ of central charge 24 with non-zero weight-one space V 1 $V_1$ is uniquely determined by the Lie algebra structure of V 1 $V_1$ .