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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.06133 |
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| _version_ | 1866929497443401728 |
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| author | Brown, Gavin Wemyss, Michael |
| author_facet | Brown, Gavin Wemyss, Michael |
| contents | This paper determines the full derived deformation theory of certain smooth rational curves C in Calabi-Yau 3-folds, by determining all higher A_\infty-products in its controlling DG-algebra. This geometric setup includes very general cases where C does not contract, cases where the curve neighbourhood is not rational, all known simple smooth 3-fold flops, and all known divisorial contractions to curves. As a corollary, it is shown that the noncommutative deformation theory of C can be described as a superpotential algebra derived from what we call free necklace polynomials, which are elements in the free algebra obtained via a closed formula from combinatorial gluing data. The description of these polynomials, together with the above results, establishes a suitably interpreted string theory prediction due to Ferrari, Aspinwall-Katz and Curto-Morrison. Perhaps most significantly, the main results give both the language and evidence to finally formulate new contractibility conjectures for rational curves in CY 3-folds, which lift Artin's celebrated results from surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_06133 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Derived deformation theory of crepant curves Brown, Gavin Wemyss, Michael Algebraic Geometry High Energy Physics - Theory Representation Theory This paper determines the full derived deformation theory of certain smooth rational curves C in Calabi-Yau 3-folds, by determining all higher A_\infty-products in its controlling DG-algebra. This geometric setup includes very general cases where C does not contract, cases where the curve neighbourhood is not rational, all known simple smooth 3-fold flops, and all known divisorial contractions to curves. As a corollary, it is shown that the noncommutative deformation theory of C can be described as a superpotential algebra derived from what we call free necklace polynomials, which are elements in the free algebra obtained via a closed formula from combinatorial gluing data. The description of these polynomials, together with the above results, establishes a suitably interpreted string theory prediction due to Ferrari, Aspinwall-Katz and Curto-Morrison. Perhaps most significantly, the main results give both the language and evidence to finally formulate new contractibility conjectures for rational curves in CY 3-folds, which lift Artin's celebrated results from surfaces. |
| title | Derived deformation theory of crepant curves |
| topic | Algebraic Geometry High Energy Physics - Theory Representation Theory |
| url | https://arxiv.org/abs/2310.06133 |