Vertex operator algebra bundles on modular curves and their associated modular forms
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911377782734848 |
|---|---|
| author | Barake, Daniel Chuchman, Owen Franc, Cameron Mason, Geoffrey Nasserden, Brett |
| author_facet | Barake, Daniel Chuchman, Owen Franc, Cameron Mason, Geoffrey Nasserden, Brett |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10686 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vertex operator algebra bundles on modular curves and their associated modular forms Barake, Daniel Chuchman, Owen Franc, Cameron Mason, Geoffrey Nasserden, Brett Number Theory Mathematical Physics Quantum Algebra 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. |
| title | Vertex operator algebra bundles on modular curves and their associated modular forms |
| topic | Number Theory Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2601.10686 |