Microlocal theory of Legendrian links and cluster algebras

Casals, Roger¹ ; Weng, Daping¹

¹ Department of Mathematics, University of California Davis, Davis, CA, United States

Geometry & topology, Tome 28 (2024) no. 2, pp. 901-1000.

Résumé

We show the existence of quasicluster $\mathcal{𝒜}$–structures and cluster Poisson structures on moduli stacks of sheaves with singular support in the alternating strand diagram of grid plabic graphs by studying the microlocal parallel transport of sheaf quantizations of Lagrangian fillings of Legendrian links. The construction is in terms of contact and symplectic topology, showing that there exists an initial seed associated to a canonical relative Lagrangian skeleton. In particular, mutable cluster $\mathcal{𝒜}$–variables are intrinsically characterized via the symplectic topology of Lagrangian fillings in terms of dually $\mathbb{𝕃}$–compressible cycles. New ingredients are introduced throughout, including the initial weave associated to a grid plabic graph, cluster mutation along nonsquare faces of a plabic graph, possibly including lollipops, the concept of sugar-free hull, and the notion of microlocal merodromy. Finally, we prove the existence of the cluster DT transformation for shuffle graphs, constructing a contact- geometric realization and an explicit reddening sequence, and establish cluster duality for the cluster ensembles.

Détail

DOI : 10.2140/gt.2024.28.901

Keywords: Legendrian, Lagrangian, cluster algebra, microlocal, sheaves

Bibliographie

[1] D G L Allegretti, Stability conditions, cluster varieties, and Riemann–Hilbert problems from surfaces, Adv. Math. 380 (2021) | DOI

[2] D Auroux, Mirror symmetry and T–duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. 1 (2007) 51

[3] D Auroux, Special Lagrangian fibrations, wall-crossing, and mirror symmetry, from: "Geometry, analysis, and algebraic geometry : forty years of the Journal of Differential Geometry" (editors H D Cao, S T Yau), Surv. Differ. Geom. 13, International (2009) 1 | DOI

[4] A Berenstein, S Fomin, A Zelevinsky, Cluster algebras, III : Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005) 1 | DOI

[5] A Björner, F Brenti, Combinatorics of Coxeter groups, 231, Springer (2005) | DOI

[6] P P Boalch, Geometry and braiding of Stokes data ; fission and wild character varieties, Ann. of Math. 179 (2014) 301 | DOI

[7] P Boalch, Poisson varieties from Riemann surfaces, Indag. Math. 25 (2014) 872 | DOI

[8] R Casals, Lagrangian skeleta and plane curve singularities, J. Fixed Point Theory Appl. 24 (2022) | DOI

[9] R Casals, H Gao, Infinitely many Lagrangian fillings, Ann. of Math. 195 (2022) 207 | DOI

[10] R Casals, L Ng, Braid loops with infinite monodromy on the Legendrian contact DGA, J. Topol. 15 (2022) 1927 | DOI

[11] R Casals, E Zaslow, Legendrian weaves : N–graph calculus, flag moduli and applications, Geom. Topol. 26 (2022) 3589 | DOI

[12] R Casals, E Gorsky, M Gorsky, J Simental, Algebraic weaves and braid varieties, preprint (2020)

[13] R Casals, E Gorsky, M Gorsky, J Simental, Positroid links and braid varieties, preprint (2021)

[14] R Casals, E Gorsky, M Gorsky, I Le, L Shen, J Simental, Cluster structures on braid varieties, preprint (2022)

[15] H Derksen, J Weyman, A Zelevinsky, Quivers with potentials and their representations, II : Applications to cluster algebras, J. Amer. Math. Soc. 23 (2010) 749 | DOI

[16] T Ekholm, K Honda, T Kálmán, Legendrian knots and exact Lagrangian cobordisms, J. Eur. Math. Soc. 18 (2016) 2627 | DOI

[17] J B Etnyre, L L Ng, V Vértesi, Legendrian and transverse twist knots, J. Eur. Math. Soc. 15 (2013) 969 | DOI

[18] V V Fock, A B Goncharov, Cluster X–varieties, amalgamation, and Poisson–Lie groups, from: "Algebraic geometry and number theory" (editor V Ginzburg), Progr. Math. 253, Birkhäuser (2006) 27 | DOI

[19] V V Fock, A B Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. 42 (2009) 865 | DOI

[20] S Fomin, M Shapiro, D Thurston, Cluster algebras and triangulated surfaces, I : Cluster complexes, Acta Math. 201 (2008) 83 | DOI

[21] S Fomin, P Pylyavskyy, E Shustin, D Thurston, Morsifications and mutations, J. Lond. Math. Soc. 105 (2022) 2478 | DOI

[22] D Gaiotto, G W Moore, A Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Comm. Math. Phys. 299 (2010) 163 | DOI

[23] D Gaiotto, G W Moore, A Neitzke, Spectral networks, Ann. Henri Poincaré 14 (2013) 1643 | DOI

[24] P Galashin, T Lam, Positroid varieties and cluster algebras, Ann. Sci. Éc. Norm. Supér. 56 (2023) 859 | DOI

[25] H Gao, L Shen, D Weng, Augmentations, fillings, and clusters, preprint (2020)

[26] H Gao, L Shen, D Weng, Positive braid links with infinitely many fillings, preprint (2020)

[27] M Gekhtman, M Shapiro, A Vainshtein, Cluster algebras and Weil–Petersson forms, Duke Math. J. 127 (2005) 291 | DOI

[28] M Gekhtman, M Shapiro, A Vainshtein, Cluster algebras and Poisson geometry, 167, Amer. Math. Soc. (2010) | DOI

[29] A B Goncharov, R Kenyon, Dimers and cluster integrable systems, Ann. Sci. Éc. Norm. Supér. 46 (2013) 747 | DOI

[30] A Goncharov, L Shen, Donaldson–Thomas transformations of moduli spaces of G-local systems, Adv. Math. 327 (2018) 225 | DOI

[31] M Gross, P Hacking, S Keel, Birational geometry of cluster algebras, Algebr. Geom. 2 (2015) 137 | DOI

[32] M Gross, P Hacking, S Keel, M Kontsevich, Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018) 497 | DOI

[33] S Guillermou, Sheaves and symplectic geometry of cotangent bundles, 440, Soc. Math. France (2023) | DOI

[34] S Guillermou, P Schapira, Microlocal theory of sheaves and Tamarkin’s non displaceability theorem, from: "Homological mirror symmetry and tropical geometry" (editors R Castano-Bernard, F Catanese, M Kontsevich, T Pantev, Y Soibelman, I Zharkov), Lect. Notes Unione Mat. Ital. 15, Springer (2014) 43 | DOI

[35] S Guillermou, M Kashiwara, P Schapira, Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems, Duke Math. J. 161 (2012) 201 | DOI

[36] P Hacking, S Keel, Mirror symmetry and cluster algebras, from: "Proceedings of the International Congress of Mathematicians, II : Invited lectures" (editors B Sirakov, P N d Souza, M Viana), World Sci. (2018) 671 | DOI

[37] R Hartshorne, Algebraic geometry, 52, Springer (1977) | DOI

[38] M B Henry, D Rutherford, Ruling polynomials and augmentations over finite fields, J. Topol. 8 (2015) 1 | DOI

[39] K Iwaki, T Nakanishi, Exact WKB analysis and cluster algebras, J. Phys. A 47 (2014) | DOI

[40] K Iwaki, T Nakanishi, Exact WKB analysis and cluster algebras, II : Simple poles, orbifold points, and generalized cluster algebras, Int. Math. Res. Not. 2016 (2016) 4375 | DOI

[41] X Jin, D Treumann, Brane structures in microlocal sheaf theory, preprint (2017)

[42] M Kashiwara, P Schapira, Microlocal study of sheaves, 128, Soc. Math. France (1985) 235

[43] B Keller, Quiver mutation and combinatorial DT-invariants, preprint (2017)

[44] T Lam, D E Speyer, Cohomology of cluster varieties, I : Locally acyclic case, Algebra Number Theory 16 (2022) 179 | DOI

[45] G Muller, Locally acyclic cluster algebras, Adv. Math. 233 (2013) 207 | DOI

[46] G Muller, A = U for locally acyclic cluster algebras, Symmetry Integrability Geom. Methods Appl. 10 (2014) | DOI

[47] D Nadler, Wrapped microlocal sheaves on pairs of pants, preprint (2016)

[48] A Neitzke, Cluster-like coordinates in supersymmetric quantum field theory, Proc. Natl. Acad. Sci. USA 111 (2014) 9717 | DOI

[49] L Ng, D Rutherford, V Shende, S Sivek, The cardinality of the augmentation category of a Legendrian link, Math. Res. Lett. 24 (2017) 1845 | DOI

[50] L Ng, D Rutherford, V Shende, S Sivek, E Zaslow, Augmentations are sheaves, Geom. Topol. 24 (2020) 2149 | DOI

[51] L Polterovich, The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991) 198 | DOI

[52] A Postnikov, Total positivity, grassmannians, and networks, preprint (2006)

[53] O M Schnürer, Six operations on dg enhancements of derived categories of sheaves, Selecta Math. 24 (2018) 1805 | DOI

[54] K Serhiyenko, M Sherman-Bennett, L Williams, Cluster structures in Schubert varieties in the Grassmannian, Proc. Lond. Math. Soc. 119 (2019) 1694 | DOI

[55] L Shen, D Weng, Cyclic sieving and cluster duality of Grassmannian, Symmetry Integrability Geom. Methods Appl. 16 (2020) | DOI

[56] L Shen, D Weng, Cluster structures on double Bott–Samelson cells, Forum Math. Sigma 9 (2021) | DOI

[57] V Shende, D Treumann, H Williams, On the combinatorics of exact Lagrangian surfaces, preprint (2016)

[58] V Shende, D Treumann, E Zaslow, Legendrian knots and constructible sheaves, Invent. Math. 207 (2017) 1031 | DOI

[59] V Shende, D Treumann, H Williams, E Zaslow, Cluster varieties from Legendrian knots, Duke Math. J. 168 (2019) 2801 | DOI

[60] B Toën, M Vaquié, Moduli of objects in dg-categories, Ann. Sci. École Norm. Sup. 40 (2007) 387 | DOI

[61] M L Yau, Surgery and isotopy of Lagrangian surfaces, from: "Proceedings of the sixth international congress of Chinese mathematicians, II" (editors C S Lin, L Yang, S T Yau, J Yu), Adv. Lect. Math. 37, International (2017) 143