Chiral de Rham complex on the upper half plane and modular forms

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 introduce an $SL(2,\mathbb R)$-invariant filtration on the global sections and show that the $\Gamma$-invariants on the graded algebra is isomorphic to certain copies of modular forms. We also give an explicit formula for the lifting of modular forms to $\Omega^{ch}(\mathbb H,\Gamma)$ and compute the character formula of $\Omega^{ch}(\mathbb H,\Gamma)$. Furthermore, we show that the vertex algebra structure modifies the Rankin-Cohen bracket, and the modified bracket becomes non-zero between constant modular forms involving the Eisenstein series.