Chiral differential operators on the upper half plane and modular forms

In this paper we study the vertex operator algebra $\mathscr D^{\text{ch}}(\mathbb H,\Gamma)$ constructed from the fixed points of the chiral differential operators on the upper half plane which is holomorphic at all the cusps, under the action of the congruence subgroup $\Gamma$. To this end, we introduce an $SL(2,\mathbb R)$-invariant filtration labeled by partition pairs and study its successive quotient. We show that the successive quotient under the cuspidal condition is isomorphic to the space of modular forms. And we also give a description of the structure of $\mathscr D^{\text{ch}}(\mathbb H,\Gamma)$ and compute its character.