Representations, sheaves and Legendrian \(( 2 , m )\) torus links. (English) Zbl 1428.57011
L. Ng, D. Rutherford, V. Shende, S. Sivek and E. Zaslow [“Augmentations are sheaves”, Preprint, arXiv:1502.04939] proved equivalence of the categories \(\mathcal{A}ug_+(\Lambda, \Bbbk)\), and \(\mathcal{S}h_1(\Lambda,\Bbbk)\), Here \(\Lambda\) is a Legendrian link in \(\mathbb{R}^3\). Objects of \(\mathcal{A}ug_+(\Lambda,\Bbbk)\) are augumentations \(\epsilon:(\mathcal{A}_\Lambda,\partial_\Lambda)\to (\Bbbk,0)\) of the Chekanov-Eliashberg graded differential algebra \((\mathcal{A}_\Lambda, \partial_\Lambda)\), and objects of \(\mathcal{S}h_1(\Lambda,\Bbbk)\) are constructible sheaves of \(\Bbbk\)-modules on the plane with singular support controlled by the front projection of \(\Lambda\) with microlocal rank \(1\). The authors say that this provides a link between the disparate worlds of holomorphic curve invariants in complex geometry and homological algebra invariants.
To generalize this theorem, the authors introduce the category \(\mathcal{R}ep_n(\Lambda, \Bbbk)\) of representations \(\rho:\mathcal{A}_\Lambda\to \mathrm{Mat}_n(\Bbbk)\) (§2, cf. [loc. cit.]). Note that by definition \(\mathcal{R}ep_1(\Lambda,\Bbbk)\) is \(\mathcal{A}ug_+(\Lambda,\Bbbk)\). As the corresponding sheaf category, the category \(\mathcal{S}h_n(\Lambda,\Bbbk)\) of constructible sheaves with microsupport on \(\Lambda\) and microlocal rank \(n\) seems appropriate. As for these categories, the authors conjecture
Conjecture. (“Representations are sheaves”). Let \(\Lambda\) be a Legendrian link in \(\mathbb{R}^3\), and let \(\Bbbk\) be a field. Then for any \(n\ge 1\), there is an \(A_\infty\) equivalence of \(A_\infty\) categories \(\mathcal{R}ep_n(\Lambda,\Bbbk)\to \mathcal{S}h_n(\Lambda,\Bbbk)\).
Corollary. If \(\Lambda\) is a Legendrian link, then the cohomoloy categories \(H^\ast\mathcal{R}ep_n(\Lambda,\Bbbk)\) and \(H^\ast\mathcal{S}h_n(\Lambda,\Bbbk)\) are equivalent.
The conjecture is not proved in this paper. But the following partial result is proved.
Theorem 1. For \(m\ge 1\), let \(\Lambda_m\) be the Legendrian \((2,m)\) torus link given by the rainbow closure of the braid \(\sigma_1^m\in B_2\) (cf. Figure 1), equipped with its standard binary Maslov potential. Then the cohomology categories \(H^\ast\mathcal{R}ep_n(\Lambda,\Bbbk)\) and \(H^\ast\mathcal{S}h_n(\Lambda,\Bbbk)\) are equivalent.
The definition of the representation category \(\mathcal{R}ep_n(\Lambda,\Bbbk)\) associated to a Legendrian link \(\Lambda\) and its elementary properties as the \(A_\infty\) category are exposed in §2, following [B. Chantraine et al., in: Proceedings of the 22nd Gökova geometry-topology conference, Gökova, Turkey, May 25–29, 2015. Somerville, MA: International Press; Gökova: Gökova Geometry-Topology Conferences (GGT). 116–150 (2016; Zbl 1358.16009)] and [loc. cit.]. Further study on \(\mathcal{R}ep_n(\Lambda_m,\Bbbk)\) is proceeded in §3 when \(\Lambda_m\) is a Legendrian \((2,m)\) torus link, taking a representation of \(\Lambda_m\) in the Legendrian isotopy class to be simply the Chekanov-Eliasberg DGA for this version (cf. Figure 2). Especially, the cohomology \(H^p\mathrm{Hom}(\rho,\rho')\) of the graded \(\mathrm{Mat}_n(\Bbbk)\) space \(\mathrm{Hom}(\rho,\rho')\), \(\rho,\rho'\) are objects in \(\mathcal{R}ep_n(\Lambda,\Bbbk)\), are computed up to \(p\le 2\) (Proposition 3.3).
\(\mathcal{S}h_n(\Lambda,\Bbbk)\) is constructed in §4 following [V. Shende et al., Invent. Math. 207, No. 3, 1031–1133 (2017; Zbl 1369.57016)], after explaining microsupport of constructible sheaves and its application to symplectic and contact geometry (cf. [M. Kashiwara and P. Schapira, Sheaves on manifolds. Berlin etc.: Springer-Verlag (1990; Zbl 0709.18001), S. Guillermou et al., Duke Math. J. 161, No. 2, 201–245 (2012; Zbl 1242.53108)]).
\(H^\ast\mathcal{S}h(\Lambda,\Bbbk)\) is computed in §5. Here \(\Lambda\) is taken as the rainbow closure of the 2-braid \(\sigma_1^m\in B_2\) (cf. Figure 1). By this choice, it is shown that the space of morphisms in \(H^\ast\mathcal{S}h_n(\Lambda_m,\Bbbk)\) is given by Ext groups between objects, and the only possible nonzero groups are \(\mathrm{Ext}^p, 0\le p\le 2\) (§5.1). \(\mathrm{Ext}^0\) and \(\mathrm{Ext}^1\) and their compositions are computed in §5.3, 5.4 and 5.5. \(\mathrm{Ext}^2\) vanishes. This is crucial and proved in §6. Theorem 1 is proved combining these results and results in §3, e.g. Proposition 3.3 (§7, the last section).
To generalize this theorem, the authors introduce the category \(\mathcal{R}ep_n(\Lambda, \Bbbk)\) of representations \(\rho:\mathcal{A}_\Lambda\to \mathrm{Mat}_n(\Bbbk)\) (§2, cf. [loc. cit.]). Note that by definition \(\mathcal{R}ep_1(\Lambda,\Bbbk)\) is \(\mathcal{A}ug_+(\Lambda,\Bbbk)\). As the corresponding sheaf category, the category \(\mathcal{S}h_n(\Lambda,\Bbbk)\) of constructible sheaves with microsupport on \(\Lambda\) and microlocal rank \(n\) seems appropriate. As for these categories, the authors conjecture
Conjecture. (“Representations are sheaves”). Let \(\Lambda\) be a Legendrian link in \(\mathbb{R}^3\), and let \(\Bbbk\) be a field. Then for any \(n\ge 1\), there is an \(A_\infty\) equivalence of \(A_\infty\) categories \(\mathcal{R}ep_n(\Lambda,\Bbbk)\to \mathcal{S}h_n(\Lambda,\Bbbk)\).
Corollary. If \(\Lambda\) is a Legendrian link, then the cohomoloy categories \(H^\ast\mathcal{R}ep_n(\Lambda,\Bbbk)\) and \(H^\ast\mathcal{S}h_n(\Lambda,\Bbbk)\) are equivalent.
The conjecture is not proved in this paper. But the following partial result is proved.
Theorem 1. For \(m\ge 1\), let \(\Lambda_m\) be the Legendrian \((2,m)\) torus link given by the rainbow closure of the braid \(\sigma_1^m\in B_2\) (cf. Figure 1), equipped with its standard binary Maslov potential. Then the cohomology categories \(H^\ast\mathcal{R}ep_n(\Lambda,\Bbbk)\) and \(H^\ast\mathcal{S}h_n(\Lambda,\Bbbk)\) are equivalent.
The definition of the representation category \(\mathcal{R}ep_n(\Lambda,\Bbbk)\) associated to a Legendrian link \(\Lambda\) and its elementary properties as the \(A_\infty\) category are exposed in §2, following [B. Chantraine et al., in: Proceedings of the 22nd Gökova geometry-topology conference, Gökova, Turkey, May 25–29, 2015. Somerville, MA: International Press; Gökova: Gökova Geometry-Topology Conferences (GGT). 116–150 (2016; Zbl 1358.16009)] and [loc. cit.]. Further study on \(\mathcal{R}ep_n(\Lambda_m,\Bbbk)\) is proceeded in §3 when \(\Lambda_m\) is a Legendrian \((2,m)\) torus link, taking a representation of \(\Lambda_m\) in the Legendrian isotopy class to be simply the Chekanov-Eliasberg DGA for this version (cf. Figure 2). Especially, the cohomology \(H^p\mathrm{Hom}(\rho,\rho')\) of the graded \(\mathrm{Mat}_n(\Bbbk)\) space \(\mathrm{Hom}(\rho,\rho')\), \(\rho,\rho'\) are objects in \(\mathcal{R}ep_n(\Lambda,\Bbbk)\), are computed up to \(p\le 2\) (Proposition 3.3).
\(\mathcal{S}h_n(\Lambda,\Bbbk)\) is constructed in §4 following [V. Shende et al., Invent. Math. 207, No. 3, 1031–1133 (2017; Zbl 1369.57016)], after explaining microsupport of constructible sheaves and its application to symplectic and contact geometry (cf. [M. Kashiwara and P. Schapira, Sheaves on manifolds. Berlin etc.: Springer-Verlag (1990; Zbl 0709.18001), S. Guillermou et al., Duke Math. J. 161, No. 2, 201–245 (2012; Zbl 1242.53108)]).
\(H^\ast\mathcal{S}h(\Lambda,\Bbbk)\) is computed in §5. Here \(\Lambda\) is taken as the rainbow closure of the 2-braid \(\sigma_1^m\in B_2\) (cf. Figure 1). By this choice, it is shown that the space of morphisms in \(H^\ast\mathcal{S}h_n(\Lambda_m,\Bbbk)\) is given by Ext groups between objects, and the only possible nonzero groups are \(\mathrm{Ext}^p, 0\le p\le 2\) (§5.1). \(\mathrm{Ext}^0\) and \(\mathrm{Ext}^1\) and their compositions are computed in §5.3, 5.4 and 5.5. \(\mathrm{Ext}^2\) vanishes. This is crucial and proved in §6. Theorem 1 is proved combining these results and results in §3, e.g. Proposition 3.3 (§7, the last section).
Reviewer: Akira Asada (Takarazuka)
MSC:
| 57R17 | Symplectic and contact topology in high or arbitrary dimension |
| 53D37 | Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category |
| 53D42 | Symplectic field theory; contact homology |
| 57K10 | Knot theory |
| 14F08 | Derived categories of sheaves, dg categories, and related constructions in algebraic geometry |
| 18F99 | Categories in geometry and topology |
Keywords:
Legendrian \((2,m)\) torus link; representation category; constructible sheaf category; microlocal rank; Ext groupReferences:
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