Chiral differential operators on the upper half plane and modular forms

TL;DR

This paper explores the structure of chiral differential operators on the upper half plane, revealing their connection to modular forms and providing a detailed description and character computation of the associated vertex operator algebra.

Contribution

It introduces a new filtration on the vertex operator algebra of chiral differential operators, linking its quotients to modular forms and analyzing its structure under congruence subgroup actions.

Findings

Successive quotients are isomorphic to spaces of modular forms.

The structure of the vertex operator algebra is explicitly described.

Character of the algebra is computed.

Abstract

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.