Nils R. Scheithauer

Construction and Classification of Holomorphic Vertex Operator Algebras

Abstract We develop an orbifold theory for finite, cyclic groups acting on holomorphic vertex operator algebras. Then we show that Schellekens’ classification of V 1 {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.

Generalized Kac¿Moody algebras, automorphic forms and Conway's group I

Abstract The moonshine properties imply that the twisted denominator identities coming from the action of the monster group on the monster algebra define modular forms. In this paper, we motivate the conjecture that the action of an extension of Conways simple group Co 1 on the fake monster algebra gives rise to automorphic forms of singular weight on Grassmannians. We prove the conjecture for elements with square-free level and nontrivial fixpoint lattice.

Some constructions of modular forms for the Weil representation of SL2(ℤ)

Modular forms for the Weil representation of SL 2 (ℤ) play an important role in the theory of automorphic forms on orthogonal groups. In this paper we give some explicit constructions of these functions. As an application, we construct new examples of generalized Kac-Moody algebras whose denominator identities are holomorphic automorphic products of singular weight. They correspond naturally to the Niemeier lattices with root systems and to the Leech lattice.

Generalized Kac-Moody algebras, automorphic forms and Conway's group II

Let Γ be a genus 0 group between Γ0(N) and its normalizer in SL 2(ℝ) where N is squarefree. We construct an automorphic product on Γ × Γ and determine its sum expansions at the different cusps. We obtain many new product identities generalizing the classical product formula of the elliptic j-function due to Zagier, Borcherds and others. These results imply that the moonshine conjecture for Conways group Co 0 is true for elements of squarefree level.

Automorphic products of singular weight

We prove some new structure results for automorphic products of singular weight. First we give a simple characterisation of the Borcherds function Φ12. Second we show that holomorphic automorphic products of singular weight on lattices of prime level exist only in small signatures and we derive an explicit bound. Finally we give a complete classification of reflective automorphic products of singular weight on lattices of prime level.

Lie Algebras, Vertex Algebras, and Automorphic Forms

Generalized Kac–Moody algebras are natural generalizations of the finite dimensional simple Lie algebras. In important cases their denominator identities are automorphic forms on orthogonal groups. The generalized Kac–Moody algebras with this property can probably be classified and realized as strings moving on suitable spacetimes. In this paper we describe these ideas in more detail.

Lie Algebren und automorphe Formen

Eine Algebra mit einem solchen Produkt wird als Lie Algebra bezeichnet. Es hat sich herausgestellt, daß enge Beziehungen zwischen gewissen unendlichdimensionalen Lie Algebren und automorphen Formen bestehen. Dies sind meromorphe Funktionen, die einfache Transformationseigenschaften unter geeigneten Gruppen aufweisen. In diesem Artikel stellen wir drei Beispiele für den Zusammenhang zwischen Lie Algebren und automorphen Formen vor. Wir skizzieren Borcherds’ Beweis der moonshine conjecture. Danach formulieren wir eine analoge Vermutung für Conways Gruppe Co0. Im letzten Abschnitt beschreiben wir Klassifikationsresultate für unendlich-dimensionale Lie Algebren.