Simple Vertex Algebras Arising From Congruence Subgroups

TL;DR

This paper constructs simple vertex algebras from congruence subgroups using the chiral de Rham complex, meromorphic modular forms, and generalized Rankin-Cohen brackets, revealing new algebraic structures related to modular forms.

Contribution

It introduces a novel construction of vertex algebras from congruence subgroups and generalizes the Rankin-Cohen bracket within this framework.

Findings

Vertex algebras from congruence subgroups are simple.

Lifting formulas for meromorphic modular forms are described.

Generalization of the Rankin-Cohen bracket to meromorphic forms.

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 $\Gamma$-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.