We develop an orbifold theory for finite, cyclic groups acting on holomorphic vertex operator algebras. Then we show that Schellekens' classification of V 1 -structures of meromorphic conformal field theories of central charge 24 is a theorem on vertex operator algebras. Finally we use these results to construct some new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds. arXiv:1507.08142v3 [math.RT]

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....

The purpose of this paper is to generalize Zhu's theorem about characters of modules over a vertex operator algebra graded by integer conformal weights, to the setting of a vertex operator superalgebra graded by rational conformal weights. To recover SL 2 (Z)invariance of the characters it turns out to be necessary to consider twisted modules alongside ordinary ones. It also turns out to be necessary, in describing the space of conformal blocks in the supersymmetric case, to include certain 'odd traces' on modules alongside traces and supertraces. We prove that the set of supertrace functions, thus supplemented, spans a finite dimensional SL 2 (Z)-invariant space. We close the paper with several examples. * email: jethrovanekeren@gmail.comIf Y (ω, z) = L(z) = n∈Z L n z −n−2 then the operators L n form a representation on V of the Virasoro algebra with some central charge c. The energy operator L 0 acts semisimply. The eigenvalue of an L 0 -eigenvector u ∈ V is called its conformal weight and is denoted ∆ u . We write V k for the set of vectors of conformal weight k.A module over a VOA V is a vector space M together with a fieldassigned to each u ∈ V . These data are to satisfy certain axioms, see Definition 2.2. A positive energy V -module is anWe say that V is rational if it has finitely many irreducible positive energy modules and every positive energy V -module is a direct sum of irreducible ones. Given V as above, Zhu introduced a second VOA structure Y [u, z] = n∈Z u ([n]) z −n−1 on V , and a new Virasoro elementω ∈ V (see Definition 2.3). Let L [0] be the new energy operator and V [k] the set of vectors of conformal weight k with respect toω.

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We study modularity of the characters of a vertex (super)algebra equipped with a family of conformal structures. Along the way we introduce the notion of rationality and cofiniteness relative to such a family. We apply the results to determine modular transformations of trace functions on admissible modules over affine Kac-Moody algebras and, via BRST reduction, trace functions on regular affine W -algebras. * 1We prove the following result as Theorem 5.12. In fact we derive it from the stronger but more technical Proposition 5.11. Theorem 1.2. Let (V, ω) be a conformal vertex (super )algebra graded by integer conformal weights. Let h ∈ V be a current satisfying the OPE relation, and such that h 0 acts semisimply on V . Assume (V, ω) to be rational relative to h and cofinite relative to h, and write Irr(V, h) for the set of irreducible h-stable positive energy V -modules. Letwhere ρ is some representation of SL 2 (Z), is satisfied for all u ∈ V if it is satisfied for u = |0 .We make some remarks on the theorem and its proof. The essential idea of the proof is to apply Zhu's modularity theorem to the vertex algebra (V, ω(z)). However ω(z) equips V with noninteger conformal weights, and Zhu's theorem actually fails in this case. This situation is rectified in the reference [28], where it is shown that modular transformations map the trace functions F M to trace functions on particular twisted modules. The task becomes to relate trace functions on twisted and untwisted V -modules. This is achieved by the use of Li's shift operators ∆(u, z) (which appear explicitly in (1.2) above). The condition of relative cofiniteness is inspired by the work [7], and was used in [28].The transformation (1.2) was uncovered in the case of N = 2 superconformal vertex algebras in [13, Theorem 9.13 (b)], with h equal to the U (1) current of the N = 2 algebra. There the functions F M are shown to be flat sections of the bundle of conformal blocks over the universal elliptic curve, and (1.2) is derived from the geometry of this bundle.We also note that a result closely related to Theorem 1.2 was recently and independently obtained in [24] in the case of V rational and C 2 -cofinite (see also [26]).An important class of vertex algebras that are relatively cofinite and generically rational in the sense discussed above is afforded by the simple affine vertex algebras at admissible level.