Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on $\mathbb{C}^2$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916906461560832 |
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| author | Ho, Quoc P. Li, Penghui |
| author_facet | Ho, Quoc P. Li, Penghui |
| contents | Using a geometric argument building on our new theory of graded sheaves, we compute the categorical trace and Drinfel'd center of the (graded) finite Hecke category $\mathsf{H}_W^\mathsf{gr} = \mathsf{Ch}^b(\mathsf{SBim}_W)$ in terms of the category of (graded) unipotent character sheaves, upgrading results of Ben-Zvi-Nadler and Bezrukavninov-Finkelberg-Ostrik. In type $A$, we relate the categorical trace to the category of $2$-periodic coherent sheaves on the Hilbert schemes $\mathsf{Hilb}_n(\mathbb{C}^2)$ of points on $\mathbb{C}^2$ (equivariant with respect to the natural $\mathbb{C}^* \times \mathbb{C}^*$ action), yielding a proof of (a $2$-periodized version of) a conjecture of Gorsky-Negut-Rasmussen which relates HOMFLY-PT link homology and the spaces of global sections of certain coherent sheaves on $\mathsf{Hilb}_n(\mathbb{C}^2)$. As an important computational input, we also establish a conjecture of Gorsky-Hogancamp-Wedrich on the formality of the Hochschild homology of $\mathsf{H}_W^\mathsf{gr}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01306 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on $\mathbb{C}^2$ Ho, Quoc P. Li, Penghui Representation Theory Algebraic Geometry Geometric Topology Quantum Algebra Primary 14F08, 14A30, 57K18. Secondary 18N25 Using a geometric argument building on our new theory of graded sheaves, we compute the categorical trace and Drinfel'd center of the (graded) finite Hecke category $\mathsf{H}_W^\mathsf{gr} = \mathsf{Ch}^b(\mathsf{SBim}_W)$ in terms of the category of (graded) unipotent character sheaves, upgrading results of Ben-Zvi-Nadler and Bezrukavninov-Finkelberg-Ostrik. In type $A$, we relate the categorical trace to the category of $2$-periodic coherent sheaves on the Hilbert schemes $\mathsf{Hilb}_n(\mathbb{C}^2)$ of points on $\mathbb{C}^2$ (equivariant with respect to the natural $\mathbb{C}^* \times \mathbb{C}^*$ action), yielding a proof of (a $2$-periodized version of) a conjecture of Gorsky-Negut-Rasmussen which relates HOMFLY-PT link homology and the spaces of global sections of certain coherent sheaves on $\mathsf{Hilb}_n(\mathbb{C}^2)$. As an important computational input, we also establish a conjecture of Gorsky-Hogancamp-Wedrich on the formality of the Hochschild homology of $\mathsf{H}_W^\mathsf{gr}$. |
| title | Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on $\mathbb{C}^2$ |
| topic | Representation Theory Algebraic Geometry Geometric Topology Quantum Algebra Primary 14F08, 14A30, 57K18. Secondary 18N25 |
| url | https://arxiv.org/abs/2305.01306 |