Dimension Formulae in Genus Zero and Uniqueness of Vertex Operator Algebras

Abstract

We prove a dimension formula for orbifold vertex operator algebras of central charge 24 by automorphisms of order n such that Γ 0 (n) is a genus zero group. We then use this formula together with the inverse orbifold construction for automorphisms of orders 2, 4, 5, 6 and 8 to establish that each of the following fifteen Lie algebras is the weight-one space V 1 of exactly one holomorphic, C 2 -cofinite vertex operator algebra V of CFT-type and central charge 24: 3,4 , corresponding to the cases 44, 33, 36, 62, 26, 52, 22, 13, 53, 48, 9, 40, 56, 21 and 7 on Schellekens' list.The strategy we use to prove uniqueness is developed in [LS16c] and is based in an essential way on the inverse orbifold construction, first developed in [EMS15] (and called "reverse orbifold" in [LS16c]). In a nutshell, the idea is as follows: Let V be a strongly rational, holomorphic vertex operator algebra of central charge 24 with weight-one Lie algebra V 1 . Any inner automorphism σ = e (2πi) adv of V 1 , v ∈ V 1 , extends to an automorphism of V . Choose such an automorphism, and suppose that V orb(σ) is isomorphic to the lattice vertex operator algebra V L . By virtue of its construction as an orbifold, V orb(σ) carries an inverse orbifold automorphism ζ of the same order as σ with V = (V orb(σ) ) orb(ζ) . Under favourable circumstances the order and the fixed-point subalgebra (Outline. We assume that the reader is familiar with vertex operator algebras and their representation theory and with the specific examples provided by lattice vertex operator algebras and affine vertex operator algebras (see, e.g. [LL04]).The article is organised as follows: In Section 2 we describe Lie algebras occurring as weight-one spaces of vertex operator algebras, inner automorphisms and the classification of the weight-one structures of strongly rational, holomorphic vertex operator algebras of central charge 24, dubbed Schellekens' list.Section 3 reviews the cyclic orbifold construction of strongly rational, holomorphic vertex operator algebras and the inverse orbifold construction. In Section 4 a genus zero dimension formula for the weight-one space of these orbifolds for central charge 24 is derived.In Section 5 the general procedure to prove the uniqueness of certain vertex operator algebras on Schellekens' list is described. This is split up into three parts, which are described in detail in Sections 6, 7 and 8 for the fifteen cases considered in this text.Finally, in Section 9 we state the main result of this paper.Acknowledgements. The authors would like to thank Ching Hung Lam and Hiroki Shimakura for helpful discussions. The second and third author both were partially supported by the DFG project "Infinite-dimensional Lie algebras in string theory". A part of the work was done while the first and second author visited the ESI in Vienna for the conference "Geometry and Representation Theory" in 2017.They are grateful to both the ESI and to the organisers of the conference. We thank the two anonymous referees for their comments and suggestions....

On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras

“…The proof of this result and of the extension of the result in [30] for all N such that the modular curve X 0 (N) has genus 0, i.e. N ∈ {2, ..., 10, 12, 13, 16, 18, 25}, is essentially obtained by writing the character ch V G explicitly in terms of the Hauptmodul for the group Γ 0 (N).…”

“…In the context of classifying holomorphic, strongly rational VOAs of central charge 24 (see Section 2.1 for definitions of these terms) and proving the "completeness" of a list of 71 Lie algebras devised by Schellekens [54], which is now known to contain every possible V 1 -space of such a VOA by work of various people (see for instance the introduction of [30] for references), van Ekeren, Möller, and Scheithauer [30] find a dimension formula for orbifold VOAs of central charge 24. A special case of their formula had previously been established by Möller in his thesis [48] and for the sake of simplicity, we only give this special case here.…”

“…In [30,Proposition 3.6], they give the following general identity for this function F a which is essential in establishing the dimension formulas.…”

71 holomorphic vertex operator algebras of central charge 24

“…Remark 6.10. After submission of this article, by using the similar techniques as in Theorem 6.5, the uniqueness of holomorphic VOAs of central charge 24 has been proved for 13, 5, 3, and 6 cases in [23], [38], [32], and [39], respectively. Therefore, the uniqueness for 70 of the 71 cases have been established.…”

Lattices, vertex algebras, and modular categories

“…In this note we give an account of the theory of Z/n-orbifolds of holomorphic vertex algebras developed by S. Möller, N. Scheithauer, and the author [62] [63], along with background material on lattices, vertex algebras and tensor categories. We discuss in particular a subtle invariant of self-dual modules over a group or algebra known as the Schur indicator.…”