高亏格黎曼曲面上的顶点算子代数丛与富克斯群的自守形式
Vertex operator algebra bundles on Riemann surfaces of higher genus and automorphic forms for Fuchsian groups
- Institute of Mathematics, Czech Academy of Science（捷克科学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该研究将顶点算子代数丛及相关自守形式构造从椭圆模曲线推广到富克斯群，针对全纯权2拟自守生成元\(E_2^\Gamma\)存在性进行拓扑二分法研究，通过多种方法给出不同情况下的结果及维数公式等。
中文摘要 AI 辅助
我们将顶点算子代数（VOA）丛及其相关自守形式的几何构造从椭圆模曲线推广到任意富克斯群\(\Gamma\subset\mathrm{PSL}_2(\mathbb{R})\)。关于全纯权\(2\)拟自守生成元\(E_2^\Gamma\)的存在出现了明显的拓扑二分法。当\(\Gamma\)有一个尖点时，我们通过抛物艾森斯坦级数的解析延拓构造\(E_2^\Gamma\)，并证明拟自守形式空间是一个自由多项式扩张，允许建立亏格一理论的代数设置，包括拟VOA结构和通过降算子对严格自守形式的刻画。相反，当\(\Gamma\)是亏格\(g\geq2\)的紧共形群时，阿蒂亚关于全纯联络的定理严格阻碍了\(E_2^\Gamma\)的存在。对于这个受阻情况，我们提供了精确的维数公式，将提升拟自守形式的不足直接与VOA内的拟原性失败联系起来，完全解决了无挠情况，并推测了对具有椭圆点的群的扩展。
英文摘要
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.
