Cohomology ring of unitary $N=(2,2)$ full vertex algebra and mirror symmetry
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913063466172416 |
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| author | Moriwaki, Yuto |
| author_facet | Moriwaki, Yuto |
| contents | We formulate two-dimensional $N=(2,2)$ supersymmetric conformal field theories in terms of unitary full vertex operator superalgebras and develop their cohomology theory. Cohomology rings, Hodge numbers, and the Witten index of a unitary $N=(2,2)$ full VOA are introduced. Using generalized full vertex operator superalgebras, spectral flow is constructed algebraically. Its periodicities are proved to be equivalent to the existence of top-degree cohomology classes, namely volume forms and holomorphic volume forms, and these characterizations yield Poincaré duality, T-duality, and Frobenius algebra structures on the cohomology rings, and thus two-dimensional topological field theories. A mirror construction for full VOAs and its relation to Hodge-theoretic mirror symmetry are also discussed. Finally, examples arising from abelian varieties, a special K3 surface, and a Landau-Ginzburg model are examined. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09919 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohomology ring of unitary $N=(2,2)$ full vertex algebra and mirror symmetry Moriwaki, Yuto Representation Theory Mathematical Physics Algebraic Geometry Complex Variables Quantum Algebra We formulate two-dimensional $N=(2,2)$ supersymmetric conformal field theories in terms of unitary full vertex operator superalgebras and develop their cohomology theory. Cohomology rings, Hodge numbers, and the Witten index of a unitary $N=(2,2)$ full VOA are introduced. Using generalized full vertex operator superalgebras, spectral flow is constructed algebraically. Its periodicities are proved to be equivalent to the existence of top-degree cohomology classes, namely volume forms and holomorphic volume forms, and these characterizations yield Poincaré duality, T-duality, and Frobenius algebra structures on the cohomology rings, and thus two-dimensional topological field theories. A mirror construction for full VOAs and its relation to Hodge-theoretic mirror symmetry are also discussed. Finally, examples arising from abelian varieties, a special K3 surface, and a Landau-Ginzburg model are examined. |
| title | Cohomology ring of unitary $N=(2,2)$ full vertex algebra and mirror symmetry |
| topic | Representation Theory Mathematical Physics Algebraic Geometry Complex Variables Quantum Algebra |
| url | https://arxiv.org/abs/2504.09919 |