Abstract
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.
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Acknowledgements
The author would like to thank Yi-Zhi Huang for his long-term support and patient guidance. The author would also like to thank Robert Bryant for his guidance on holonomy groups over the tensor powers of the tangent bundle and his correction on the statement of Lemma 3.5. The author would also like to thank Igor Frenkel, Nicholas Lai, Eric Schippers, and Nolan Wallach for helpful discussion.
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Qi, F. Meromorphic open-string vertex algebras and modules over two-dimensional orientable space forms. Lett Math Phys 111, 27 (2021). https://doi.org/10.1007/s11005-021-01365-6
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DOI: https://doi.org/10.1007/s11005-021-01365-6
Keywords
- Noncommutative vertex operators
- Space forms
- Parallel tensors
- Covariant derivatives
- Eigenfunctions
- Classification of irreducible modules