Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866916146021662720 |
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| author | Gui, Bin |
| author_facet | Gui, Bin |
| contents | A unitary and strongly rational vertex operator algebra (VOA) $V$ is called strongly unitary if all irreducible $V$-modules are unitarizable. A strongly unitary VOA $V$ is called completely unitary if for each unitary $V$-modules $W_1$, $W_2$ the canonical nondegenerate Hermitian form on the fusion product $W_1\boxtimes W_2$ is positive. It is known that if $V$ is completely unitary, then the modular category of unitary $V$-modules is unitary [Gui19b], and all simple VOA extensions of V are automatically unitary and moreover completely unitary [Gui22, CGGH23].
In this paper, we give a geometric characterization of the positivity of the Hermitian product on $W_1$ and $W_2$, which helps us prove that the positivity is always true when the fusion product $W_1\boxtimes W_2$ is an irreducible and unitarizable $V$-module. We give several applications: (1) We show that if $V$ is a unitary (strongly rational) holomorphic VOA with a finite cyclic unitary automorphism group $G$, and if $V^G$ is strongly unitary, then $V^G$ is completely unitary. This result applies to the cyclic permutation orbifolds of unitary holomophic VOAs. (2) We show that if $V$ is unitary and strongly rational, and if $U$ is a simple current extension which is unitarizable as a $V$-module, then $U$ is a unitary VOA. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11856 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules Gui, Bin Quantum Algebra Mathematical Physics Representation Theory A unitary and strongly rational vertex operator algebra (VOA) $V$ is called strongly unitary if all irreducible $V$-modules are unitarizable. A strongly unitary VOA $V$ is called completely unitary if for each unitary $V$-modules $W_1$, $W_2$ the canonical nondegenerate Hermitian form on the fusion product $W_1\boxtimes W_2$ is positive. It is known that if $V$ is completely unitary, then the modular category of unitary $V$-modules is unitary [Gui19b], and all simple VOA extensions of V are automatically unitary and moreover completely unitary [Gui22, CGGH23]. In this paper, we give a geometric characterization of the positivity of the Hermitian product on $W_1$ and $W_2$, which helps us prove that the positivity is always true when the fusion product $W_1\boxtimes W_2$ is an irreducible and unitarizable $V$-module. We give several applications: (1) We show that if $V$ is a unitary (strongly rational) holomorphic VOA with a finite cyclic unitary automorphism group $G$, and if $V^G$ is strongly unitary, then $V^G$ is completely unitary. This result applies to the cyclic permutation orbifolds of unitary holomophic VOAs. (2) We show that if $V$ is unitary and strongly rational, and if $U$ is a simple current extension which is unitarizable as a $V$-module, then $U$ is a unitary VOA. |
| title | Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules |
| topic | Quantum Algebra Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2306.11856 |