Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras II: Convergence of Sewing and Higher Genus Pseudo-$q$-traces
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915475824312320 |
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| author | Gui, Bin Zhang, Hao |
| author_facet | Gui, Bin Zhang, Hao |
| contents | Let $\mathbb V=\bigoplus_{n\in\mathbb N}\mathbb V(n)$ be a $C_2$-cofinite vertex operator algebra. We prove the convergence of Segal's sewing of conformal blocks associated to analytic families of pointed compact Riemann surfaces and grading-restricted generalized $\mathbb V^{\otimes N}$-modules (where $N=1,2,\dots$) that are not necessarily tensor products of $\mathbb V$-modules, generalizing significantly the results on convergence in [Gui24].
We show that ``higher genus pseudo-$q$-traces" (called pseudo-sewing in this article) can be recovered from the above generalization of Segal's sewing to $\mathbb{V}^{\otimes N}$-modules. Therefore, our result on the convergence of the generalized Segal's sewing implies the convergence of pseudo-sewing, and hence covers both the convergence of genus-$0$ sewing in [Hua05a,HLZ12] and the convergence of pseudo-$q$-traces in [Miy04] and [Fio16].
Using a similar method, we also prove the convergence of Virasoro uniformization, i.e., the convergence of conformal blocks deformed by non-automomous meromorphic vector fields near the marked points. The local freeness of the analytic sheaves of conformal blocks is a consequence of this convergence. It will be used in the third paper of this series to prove the sewing-factorization theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07707 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras II: Convergence of Sewing and Higher Genus Pseudo-$q$-traces Gui, Bin Zhang, Hao Quantum Algebra Mathematical Physics Representation Theory Let $\mathbb V=\bigoplus_{n\in\mathbb N}\mathbb V(n)$ be a $C_2$-cofinite vertex operator algebra. We prove the convergence of Segal's sewing of conformal blocks associated to analytic families of pointed compact Riemann surfaces and grading-restricted generalized $\mathbb V^{\otimes N}$-modules (where $N=1,2,\dots$) that are not necessarily tensor products of $\mathbb V$-modules, generalizing significantly the results on convergence in [Gui24]. We show that ``higher genus pseudo-$q$-traces" (called pseudo-sewing in this article) can be recovered from the above generalization of Segal's sewing to $\mathbb{V}^{\otimes N}$-modules. Therefore, our result on the convergence of the generalized Segal's sewing implies the convergence of pseudo-sewing, and hence covers both the convergence of genus-$0$ sewing in [Hua05a,HLZ12] and the convergence of pseudo-$q$-traces in [Miy04] and [Fio16]. Using a similar method, we also prove the convergence of Virasoro uniformization, i.e., the convergence of conformal blocks deformed by non-automomous meromorphic vector fields near the marked points. The local freeness of the analytic sheaves of conformal blocks is a consequence of this convergence. It will be used in the third paper of this series to prove the sewing-factorization theorem. |
| title | Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras II: Convergence of Sewing and Higher Genus Pseudo-$q$-traces |
| topic | Quantum Algebra Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2411.07707 |