Chiral de Rham Complex on the Upper Half Plane and Modular Forms

For any congruence subgroup Γ, we study the vertex operator algebra Ω ᶜʰ( H,Γ ) constructed from the Γ-invariant global sections of the chiral de Rham complex on the upper half plane, which are holomorphic at all the cusps. We construct a basis of Ω ᶜʰ( H,Γ ) in terms of modular forms of Γ and compute its character. We show that the vertex operations on Ω ᶜʰ( H,Γ ) are determined by a modification of the Rankin–Cohen brackets of modular forms.