Vertex operator algebra bundles on Riemann surfaces of higher genus and automorphic forms for Fuchsian groups
Year: 2026
Type: preprint
Abstract: We generalize the geometric construction of vertex operator algebra (VOA) bundles and their associated automorphic forms from the elliptic modular curve to arbitrary Fuchsian groups $ Γ\subset \mathrm{PSL}_2(\mathbb{R})$. A sharp topological dichotomy emerges regarding the existence of a holomorphic weight-$2$ quasi-automorphic generator $E_2^Γ$. When $Γ$ has a cusp, we construct $E_2^Γ$ via the analytic continuation of parabolic Eisenstein series and prove that the space of quasi-automorphic forms is a free polynomial extension, allowing the algebraic setup of the genus one theory, including the quasi-VOA structure and the characterization of strict automorphic forms via a lowering operator. Conversely, when $Γ$ is cocompact of genus $g\ge 2$, Atiyah theorem on holomorphic connections rigorously obstructs the existence of $E_2^Γ$. For this obstructed case, we provide dimension formulas that link the shortage in lifting quasi-automorphic forms to the failure of quasi-primarity within the VOA, fully resolving the torsion-free case and conjecturing the extension to groups with elliptic points.
Source: arXiv (Cornell University)
Author: Zuevsky, A.
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Akademie Věd České Republiky
Awards: G1741043573, G3967893287
OpenAlex ID: W7168267231