We study the meromorphic open-string vertex algebras and their modules over the two-dimensional Riemannian manifolds that are complete, connected, orientable, and of constant sectional curvature \(K\ne 0\). Using the parallel tensors, we explicitly determine a basis for the meromorphic open-string vertex algebra, its module generated by an eigenfunction for the Laplace–Beltrami operator, and its irreducible quotient. We also study the modules generated by the lowest weight subspace satisfying a geometrically interesting condition. It is showed that every irreducible module of this type is generated by some (local) eigenfunction on the manifold. A classification is given for modules of this type admitting a composition series of finite length. In particular and remarkably, if every composition factor is generated by eigenfunctions of special eigenvalue \(p(p-1)K\) for some \(p\in {\mathbb {Z}}_+\), then the module is completely reducible.

我们研究完备的，连通的，可定向的，且具有恒定截面曲率\（K \ ne 0 \）的二维黎曼流形上的亚纯开放字符串顶点代数及其模块。使用平行张量，我们明确确定亚纯开弦顶点代数的基础，由Laplace–Beltrami算符的本征函数生成的模块及其不可约商。我们还研究了满足几何有趣条件的最低权重子空间生成的模块。结果表明，该类型的每个不可约模块都是由流形上的某些（局部）本征函数生成的。对于这种类型的模块，给出了允许有限长度组成序列的分类。特别值得注意的是，如果每个构成因子都是由特殊特征值\（p（p-1）K \）的特征函数生成的，对于某些\（p \ in {\ mathbb {Z}} _ + \），则该模块为完全可还原。