Modularity of Vertex Operator Algebra Correlators with Zero Modes

TL;DR

This paper studies how vertex operator algebra correlators with zero modes transform under modular transformations, revealing they behave like quasi-modular or quasi-Jacobi forms, and introduces related algebraic structures.

Contribution

It establishes the modular transformation properties of zero mode correlators and introduces higher weight field algebras with similar properties, providing a simplified proof of Miyamoto's results.

Findings

Correlators with zero modes transform as quasi-modular forms.

Mixed correlators transform as quasi-Jacobi forms.

Introduces higher weight field algebras with zero mode correlator properties.

Abstract

It is known from Zhu's results that under modular transformations, correlators of rational $C_2$-cofinite vertex operator algebras transform like Jacobi forms. We investigate the modular transformation properties of VOA correlators that have zero modes inserted. We derive recursion relations for such correlators and use them to establish modular transformation properties. We find that correlators with only zero modes transform like quasi-modular forms, and mixed correlators with both zero modes and vertex operators transform like quasi-Jacobi forms. As an application of our results, we introduce algebras of higher weight fields whose zero mode correlators mimic the properties of those of weight 1 fields. We also give a simplified proof of the weight 1 transformation properties originally proven by Miyamoto.