Genus $g$ Virasoro Correlation Functions for Vertex Operator Algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910863865151488 |
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| author | Tuite, Michael P. Welby, Michael |
| author_facet | Tuite, Michael P. Welby, Michael |
| contents | For a simple, self-dual, strong CFT-type vertex operator algebra (VOA) of central charge $c$, we describe the Virasoro $n$-point correlation function on a genus $g$ marked Riemann surface in the Schottky uniformisation. We show that this $n$-point function determines the correlation functions for all Virasoro vacuum descendants. Using our recent work on genus $g$ Zhu recursion, we show that the Virasoro $n$-point function is determined by a differential operator $\mathcal{D}_{n}$ acting on the genus $g$ VOA partition function normalised by the Heisenberg partition function to the power of $c$. We express $\mathcal{D}_{n}$ as the sum of weights over certain Virasoro graphs where the weights explicitly depend on $c$, the classical bidifferential of the second kind, the projective connection, holomorphic 1-forms and derivatives with respect to any $3g-3$ locally independent period matrix elements. We also describe the modular properties of $\mathcal{D}_{n}$ under a homology base change. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05553 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Genus $g$ Virasoro Correlation Functions for Vertex Operator Algebras Tuite, Michael P. Welby, Michael Quantum Algebra High Energy Physics - Theory 17B69, 30F10, 32G15 For a simple, self-dual, strong CFT-type vertex operator algebra (VOA) of central charge $c$, we describe the Virasoro $n$-point correlation function on a genus $g$ marked Riemann surface in the Schottky uniformisation. We show that this $n$-point function determines the correlation functions for all Virasoro vacuum descendants. Using our recent work on genus $g$ Zhu recursion, we show that the Virasoro $n$-point function is determined by a differential operator $\mathcal{D}_{n}$ acting on the genus $g$ VOA partition function normalised by the Heisenberg partition function to the power of $c$. We express $\mathcal{D}_{n}$ as the sum of weights over certain Virasoro graphs where the weights explicitly depend on $c$, the classical bidifferential of the second kind, the projective connection, holomorphic 1-forms and derivatives with respect to any $3g-3$ locally independent period matrix elements. We also describe the modular properties of $\mathcal{D}_{n}$ under a homology base change. |
| title | Genus $g$ Virasoro Correlation Functions for Vertex Operator Algebras |
| topic | Quantum Algebra High Energy Physics - Theory 17B69, 30F10, 32G15 |
| url | https://arxiv.org/abs/2503.05553 |