Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves
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arXiv
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866917877549891584 |
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| author | Aganagic, Mina |
| author_facet | Aganagic, Mina |
| contents | We derive two geometric approaches to categorification of quantum invariants of links associated to an arbitrary compact simple Lie group $^L{G}$. In part I, we describe the first approach, based on an equivariant derived category of coherent sheaves on ${\cal X}$, the moduli space of singular $G$-monopoles, where $G$ is related to $^LG$ by Langlands duality. In part II, we describe the second approach, based on the derived category of a Fukaya-Seidel category of a Calabi-Yau $Y$ with potential $W$. The two approaches are related by a version of mirror symmetry, which plays a crucial role in the story. In part III, we explain the string theory origin of these results, and the relation to an approach due to Witten. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2004_14518 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves Aganagic, Mina High Energy Physics - Theory Algebraic Geometry Representation Theory Symplectic Geometry We derive two geometric approaches to categorification of quantum invariants of links associated to an arbitrary compact simple Lie group $^L{G}$. In part I, we describe the first approach, based on an equivariant derived category of coherent sheaves on ${\cal X}$, the moduli space of singular $G$-monopoles, where $G$ is related to $^LG$ by Langlands duality. In part II, we describe the second approach, based on the derived category of a Fukaya-Seidel category of a Calabi-Yau $Y$ with potential $W$. The two approaches are related by a version of mirror symmetry, which plays a crucial role in the story. In part III, we explain the string theory origin of these results, and the relation to an approach due to Witten. |
| title | Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves |
| topic | High Energy Physics - Theory Algebraic Geometry Representation Theory Symplectic Geometry |
| url | https://arxiv.org/abs/2004.14518 |