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]

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We prove a dimension formula for the weight-1 subspace of a vertex operator algebra V orb(g) obtained by orbifolding a strongly rational, holomorphic vertex operator algebra V of central charge 24 with a finite order automorphism g. Based on an upper bound derived from this formula we introduce the notion of a generalised deep hole in Aut(V ).We then give a construction of all 70 strongly rational, holomorphic vertex operator algebras of central charge 24 with non-vanishing weight-1 space as orbifolds of the Leech lattice vertex operator algebra V Λ associated with generalised deep holes. This provides the first uniform construction of these vertex operator algebras and naturally generalises the construction of the 23 Niemeier lattices with non-vanishing root system from the deep holes of the Leech lattice Λ by Conway, Parker and Sloane.Contents 42 References 46

We show that the physical states of a 10 dimensional superstring moving on a torus form a generalized Kac Moody superalgebra. This gives the first explicit realizations of these algebras. For a special torus the denominator function of this algebra is an automorphic form so that we can determine the simple roots. We call this algebra the fake monster superalgebra. Academic Press