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

Abstract

Abstract For any congruence subgroup $\Gamma $, we study the vertex operator algebra $\Omega ^{ch}(\mathbb H,\Gamma )$ constructed from the $\Gamma $-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 $\Omega ^{ch}(\mathbb H,\Gamma )$ in terms of modular forms of $\Gamma $ and compute its character. We show that the vertex operations on $\Omega ^{ch}(\mathbb H,\Gamma )$ are determined by a modification of the Rankin–Cohen brackets of modular forms.