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

TL;DR

This paper explores the structure of a vertex operator algebra derived from the chiral de Rham complex on the upper half plane, revealing deep connections with modular forms, their lifts, and algebraic brackets.

Contribution

It introduces a new $SL(2,\mathbb R)$-invariant filtration, relates invariants to modular forms, and provides explicit formulas for lifts and character computations.

Findings

Invariant graded algebra is isomorphic to modular forms

Explicit formula for lifting modular forms to the vertex algebra

Modified Rankin-Cohen bracket involving Eisenstein series

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.