Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras I: Propagation and Dual Fusion Products
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| Format: | Preprint |
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2023
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| _version_ | 1866908290804350976 |
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| author | Gui, Bin Zhang, Hao |
| author_facet | Gui, Bin Zhang, Hao |
| contents | This is the first paper of a three-part series in which we develop a theory of conformal blocks for $C_2$-cofinite vertex operator algebras (VOAs) that are not necessarily rational. The ultimate goal of this series is to prove a sewing-factorization theorem (and in particular, a factorization formula) for conformal blocks over holomorphic families of compact Riemann surfaces, associated to grading-restricted (generalized) modules of $C_2$-cofinite VOAs.
In this paper, we prove that if $\mathbb V$ is a $C_2$-cofinite VOA, if $\mathfrak X$ is a compact Riemann surface with $N$ incoming marked points and $M$ outgoing ones, each equipped with a local coordinate, and if $\mathbb W$ is a grading-restricted $\mathbb V^{\otimes N}$-modules, then the ``dual fusion product" exists as a grading-restricted $\mathbb V^{\otimes M}$-module. Indeed, we prove a more general version of this result without assuming $\mathbb V$ to be $C_2$-cofinite. Our main method is a generalization of the propagation of conformal blocks. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_10180 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras I: Propagation and Dual Fusion Products Gui, Bin Zhang, Hao Quantum Algebra Mathematical Physics Representation Theory This is the first paper of a three-part series in which we develop a theory of conformal blocks for $C_2$-cofinite vertex operator algebras (VOAs) that are not necessarily rational. The ultimate goal of this series is to prove a sewing-factorization theorem (and in particular, a factorization formula) for conformal blocks over holomorphic families of compact Riemann surfaces, associated to grading-restricted (generalized) modules of $C_2$-cofinite VOAs. In this paper, we prove that if $\mathbb V$ is a $C_2$-cofinite VOA, if $\mathfrak X$ is a compact Riemann surface with $N$ incoming marked points and $M$ outgoing ones, each equipped with a local coordinate, and if $\mathbb W$ is a grading-restricted $\mathbb V^{\otimes N}$-modules, then the ``dual fusion product" exists as a grading-restricted $\mathbb V^{\otimes M}$-module. Indeed, we prove a more general version of this result without assuming $\mathbb V$ to be $C_2$-cofinite. Our main method is a generalization of the propagation of conformal blocks. |
| title | Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras I: Propagation and Dual Fusion Products |
| topic | Quantum Algebra Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2305.10180 |