Niemeier Lattices and Holomorphic VOAs of Central Charge 24
We prove a systematic construction of all 70 strongly rational,
holomorphic VOAs $V$ of central charge 24 with non-zero weight-one space
$V_1$ as cyclic orbifold constructions associated with the 24 Niemeier
lattice VOAs $V_N$ and certain 230 "good" automorphisms of small order
We show that up to algebraic conjugacy these automorphisms are exactly
the generalised deep holes, as introduced in [Möller-Scheithauer-2019],
of the Niemeier lattice VOAs 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-Scheithauer-2019] this gives three different *uniform*
constructions of these VOAs, which are however related through 11
algebraic conjugacy classes in $\Co_0$.
Finally, by considering the inverse orbifold constructions associated
with the 230 "good" automorphisms, we give the first *uniform* proof of
the fact that each strongly rational, holomorphic VOA $V$ of central
charge 24 with non-zero weight-one space $V_1$ is uniquely determined by
the Lie algebra structure of $V_1$.
This is joint work with Gerald Höhn.