Vertex operator algebra bundles on modular curves and their associated modular forms

TL;DR

The paper builds a geometric framework tying vertex operator algebras to vector bundles on the modular curve X(1), formulating V-valued modular and quasi-modular forms through a doubled QVOA structure Q(V) ≅ Q ⊗ V^{(2)}. It proves that modular forms M(V) sit inside Q(V) as the kernel of a lowering operator Λ = (12/2πi) ∂/∂E_2 ⊗ Id + Id ⊗ L(1), and provides an explicit M-linear isomorphism M ⊗ V^{(2)} ≅ M(V), yielding a concrete description of the associated VOA bundle. The work develops the algebraic machinery of Q(V) as a QVOA, introduces lowering and weight-preserving operators (Λ and P), analyzes dimensions and decompositions, and shows that Hecke eigenstructures for M(V) align with those of classical modular forms. The framework allows transferring the classical theory of modular and quasi-modular forms, including Hecke theory and L-functions, to VOA-valued objects, enabling systematic study of VOA characters through automorphic techniques. This fusion of VOA geometry with modular/quasi-modular theory provides a robust toolkit for understanding VOA characters in modular settings and connects to broader arithmetic-geometric contexts via the Hecke action on M(V) and Q(V).

Abstract

This paper describes the vector bundle on the elliptic modular curve that is associated to a vertex operator algebra $V$ (VOA) or more generally a quasi-vertex operator algebra (QVOA), with a view towards future applications aimed at studying the characters of VOAs. We explain how the modes of sections of $V$ give rise naturally to $V$-valued quasi-modular forms. The space $Q(V)$ of $V$-valued quasi-modular forms is endowed with the structure of a doubled QVOA, and in particular the algebra $Q$ of quasi-modular forms is itself a doubled QVOA. $Q(V)$ also admits a natural derivative operator arising from the connection on the bundle defined by $V$ and the modular derivative, which we call the raising operator. We introduce an associated lowering operator $Λ$ on $Q(V)$ having the property that the $V$-valued modular forms $M(V)\subseteq Q(V)$ are the kernel of $Λ$. This extends the classical theory of scalar-valued quasi-modular forms. We exhibit an explicit isomorphism of $M(V)$ with $M \otimes V$. Finally, the coordinate invariance of vertex operators implies that $M(V)$ has a natural Hecke theory, and we use this isomorphism to fully describe the Hecke eigensystems: they are the same as the systems of eigenvalues that arise from scalar-valued quasi-modular forms.

Theorems & Definitions (82)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Remark 2.2
• Remark 3.1
• Remark 3.2
• Definition 3.3
• Lemma 3.4
• proof
• proof : Elementary proof that \ref{['eq:Vaction']} is a left group action