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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.17730 |
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| _version_ | 1866918435970088960 |
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| author | Cazaux, Maxime |
| author_facet | Cazaux, Maxime |
| contents | We compute the genus-0 permutation-equivariant quantum K-theory of Fermat singularities, in parallel with the Givental-Lee theory for projective varieties. We extend Givental-Tonita's formalism of adelic Lagrangian cones to the singularity theory, and we obtain explicit $I$-functions for the invariants, which satisfy the same $q$-difference equation as Givental's $I$-function of the associated hypersurface. This can be regarded as an extension of the Landau-Ginzburg/Calabi-Yau correspondence, although a discrepancy between the two sides sides emerges in K-theory. In the case of the quintic threefold, both generating functions satisfy a $q$-difference equation of degree $25$; the hypersurface $I$-function only spans a $5$-dimensional subspace of solutions, while the singularity $I$-function spans the full space of solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17730 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Permutation-equivariant quantum K-theory of Fermat singularities Cazaux, Maxime Algebraic Geometry We compute the genus-0 permutation-equivariant quantum K-theory of Fermat singularities, in parallel with the Givental-Lee theory for projective varieties. We extend Givental-Tonita's formalism of adelic Lagrangian cones to the singularity theory, and we obtain explicit $I$-functions for the invariants, which satisfy the same $q$-difference equation as Givental's $I$-function of the associated hypersurface. This can be regarded as an extension of the Landau-Ginzburg/Calabi-Yau correspondence, although a discrepancy between the two sides sides emerges in K-theory. In the case of the quintic threefold, both generating functions satisfy a $q$-difference equation of degree $25$; the hypersurface $I$-function only spans a $5$-dimensional subspace of solutions, while the singularity $I$-function spans the full space of solutions. |
| title | Permutation-equivariant quantum K-theory of Fermat singularities |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2410.17730 |