Reflective modular forms and vertex operator algebras

In this thesis we mainly study strongly rational, holomorphic vertex operator algebras and reflective modular forms. First we associate the Lie algebra of physical states to a vertex operator algebra of central charge c=24. We study the corresponding Lie bracket as a bilinear map between weight spaces of the vertex operator algebra. This makes use of no-ghost-isomorphisms. A careful analysis of the no-ghost theorem yields methods to evaluate those bilinear maps explicitly in terms of vertex algebra operations. Then we decompose such holomorphic vertex operator algebras according to their affine substructure and show that the corresponding characters are vector-valued modular forms for a coroot lattice, suitably enriched by simple currents. The associated automorphic product yields the product side of the denominator identity of the Lie algebra of physical states. Since this is a generalized Kac-Moody algebra it follows that this automorphic product is reflective. Finally we study lattices that admit a reflective modular form. We show, that there are just finitely many such lattices of even signature, which split rescaled hyperbolic planes. We determine explicit bounds for the levels.