Schellekens' list and the very strange formula

Orbifold construction and Lorentzian construction of Leech lattice vertex operator algebra

“…Since (K 0 − N 0 )|24 and (K 0 − N 0 , 3) = 1, we have K 0 − N 0 = 1, 2, 4, 8 and so (N 0 , K 0 ) = (9, 10), (9,11), (9,13), (9,17), (15,16), (15,17), (15,19), (15,23), (21,22), (21,23), (21,25), (21,29).…”

“…They called these 69 (isomorphism classes of) automorphisms "generalized deep holes of Leech lattice". In [9], another proof that any holomorphic VOA of central charge 24 with V 1 = 0 can be constructed by a single orbifold construction from the Leech lattice is also given. Moreover, a relatively simpler proof for the Schellekens' list is obtained.…”

“…Although this theorem has already been proved in [9] and [18] by using dimensional formula, our main aim is to analyze structural relations among holomorphic VOAs in the process of the proof.…”

Automorphism groups of cyclic orbifold vertex operator algebras associated with the Leech lattice and some non-prime isometries

“…The classification of holomorphic vertex operator algebras (VOAs) of central charge 24 has been completed except for the uniqueness of the moonshine VOA, and several uniform proofs are proposed (see [Hö,MS,HM,ELMS21,CLM] and the references therein). One of them, proposed by Höhn in [Hö], is to view a holomorphic VOA V of central charge 24 with V 1 = 0 as a simple current extension of the tensor product VOA V Lg ⊗ V ĝ Λg .…”

Holomorphic integer graded vertex superalgebras

“…The first step in Schellekens' classification consisted in identifying the solutions to (1.1), which turn out to be 221 in number. The reduction to the final list of 71 relied on the solution of complicated integer programming problems (see also [31]), though recently in [30] a more conceptual approach to this reduction has been given, based on orbifolds of the Leech lattice vertex algebra and Kac's very strange formula.…”