Simple Vertex Algebras Arising From Congruence Subgroups

Abstract

Chiral de Rham complex introduced by Malikov et al. in 1998, is a sheaf of vertex algebras on any complex analytic manifold or non-singular algebraic variety. Starting from the vertex algebra of global sections of chiral de Rham complex on the upper half plane, we consider the subspace of $\hat{I}“$-invariant sections that are meromorphic at the cusps. The space is again a vertex operator algebra, with a linear basis consisting of lifting formulas of meromorphic modular forms. We will describe two types of lifting formulas, and generalize the Rankin-Cohen bracket to the meromorphic modular forms. As an application, we will show that the vertex algebras constructed by congruence subgroups are simple.

25 pages, to appear in Advances in Mathematics