Modular functors from conformal blocks of rational vertex operator algebras
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915483749449728 |
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| author | Damiolini, Chiara Woike, Lukas |
| author_facet | Damiolini, Chiara Woike, Lukas |
| contents | For a vertex operator algebra $V$, one may naturally define spaces of conformal blocks following a construction of Frenkel-Ben-Zvi generalized by Damiolini-Gibney-Tarasca. If $V$ is strongly rational, these spaces of conformal blocks form vector bundles over a suitable moduli space of algebraic curves. In this article, we establish, under the same assumptions, the widely expected topological result that the spaces of conformal blocks produce a modular functor, i.e. a modular algebra over an extension of the surface operad. This entails that the category $\mathcal{C}_V$ of admissible $V$-modules inherits from the topology of genus zero surfaces a ribbon Grothendieck-Verdier structure that leads even to the structure of a modular fusion category whose structure comes directly from the spaces of conformal blocks of $V$. As a direct consequence, we prove that the modular functor from conformal blocks extends to a three-dimensional topological field theory and comes with a description in terms of factorization homology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05845 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modular functors from conformal blocks of rational vertex operator algebras Damiolini, Chiara Woike, Lukas Quantum Algebra Mathematical Physics Algebraic Geometry Algebraic Topology For a vertex operator algebra $V$, one may naturally define spaces of conformal blocks following a construction of Frenkel-Ben-Zvi generalized by Damiolini-Gibney-Tarasca. If $V$ is strongly rational, these spaces of conformal blocks form vector bundles over a suitable moduli space of algebraic curves. In this article, we establish, under the same assumptions, the widely expected topological result that the spaces of conformal blocks produce a modular functor, i.e. a modular algebra over an extension of the surface operad. This entails that the category $\mathcal{C}_V$ of admissible $V$-modules inherits from the topology of genus zero surfaces a ribbon Grothendieck-Verdier structure that leads even to the structure of a modular fusion category whose structure comes directly from the spaces of conformal blocks of $V$. As a direct consequence, we prove that the modular functor from conformal blocks extends to a three-dimensional topological field theory and comes with a description in terms of factorization homology. |
| title | Modular functors from conformal blocks of rational vertex operator algebras |
| topic | Quantum Algebra Mathematical Physics Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2507.05845 |