Vertex operator algebra bundles on Riemann surfaces of higher genus and automorphic forms for Fuchsian groups

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 $\Gamma \subset {\mathrm{PSL}}_2(\mathbb{R})$. A sharp topological dichotomy emerges regarding the existence of a holomorphic weight-$2$ quasi-automorphic generator $E_2^{\Gamma }$. When $\Gamma$ has a cusp, we construct $E_2^{\Gamma }$ 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 $\Gamma$ is cocompact of genus $g\ge 2$, Atiyah's theorem on holomorphic connections rigorously obstructs the existence of $E_2^{\Gamma }$. For this obstructed case, we provide exact dimension formulas that link the shortage in lifting quasi-automorphic forms directly to the failure of quasi-primarity within the VOA, fully resolving the torsion-free case and conjecturing the extension to groups with elliptic points.