Quivers, Floer cohomology, and braid group actions

Khovanov, Mikhail¹ ; Seidel, Paul²

¹ Department of Mathematics, University of California at Davis, Davis, California 95616-8633
² Department of Mathematics, Ecole Polytechnique, F-91128 Palaiseau, France – and – School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540

Journal of the American Mathematical Society, Tome 15 (2002) no. 1, pp. 203-271

Résumé

We consider the derived categories of modules over a certain family $A_m$ ($m \geq 1$) of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which are the Milnor fibres of simple singularities of type $A_m.$ We show that each of these two rather different objects encodes the topology of curves on an $(m+1)$-punctured disc. We prove that the braid group $B_{m+1}$ acts faithfully on the derived category of $A_m$-modules, and that it injects into the symplectic mapping class group of the Milnor fibers. The philosophy behind our results is as follows. Using Floer cohomology, one should be able to associate to the Milnor fibre a triangulated category (its construction has not been carried out in detail yet). This triangulated category should contain a full subcategory which is equivalent, up to a slight difference in the grading, to the derived category of $A_m$-modules. The full embedding would connect the two occurrences of the braid group, thus explaining the similarity between them.

Détail

DOI : 10.1090/S0894-0347-01-00374-5

Bibliographie

[1] Arnol′D, V. I. Some remarks on symplectic monodromy of Milnor fibrations 1995 99 103

[2] Bernšteĭn, I. N., Gel′Fand, I. M., Gel′Fand, S. I. Algebraic vector bundles on 𝑃ⁿ and problems of linear algebra Funktsional. Anal. i Prilozhen. 1978 66 67

[3] Beilinson, Alexander, Ginzburg, Victor, Soergel, Wolfgang Koszul duality patterns in representation theory J. Amer. Math. Soc. 1996 473 527

[4] Bernstein, Joseph, Frenkel, Igor, Khovanov, Mikhail A categorification of the Temperley-Lieb algebra and Schur quotients of 𝑈(𝔰𝔩₂) via projective and Zuckerman functors Selecta Math. (N.S.) 1999 199 241

[5] Bernstein, J. N., Gel′Fand, S. I. Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras Compositio Math. 1980 245 285

[6] Robertson, M. S. The variation of the sign of 𝑉 for an analytic function 𝑈+𝑖𝑉 Duke Math. J. 1939 512 519

[7] Birman, Joan S., Hilden, Hugh M. On isotopies of homeomorphisms of Riemann surfaces Ann. of Math. (2) 1973 424 439

[8] Brieskorn, Egbert Über die Auflösung gewisser Singularitäten von holomorphen Abbildungen Math. Ann. 1966 76 102

[9] Eliashberg, Y., Hofer, H., Salamon, D. Lagrangian intersections in contact geometry Geom. Funct. Anal. 1995 244 269

[10] Floer, Andreas Morse theory for Lagrangian intersections J. Differential Geom. 1988 513 547

[11] Floer, Andreas A relative Morse index for the symplectic action Comm. Pure Appl. Math. 1988 393 407

[12] Floer, Andreas Witten’s complex and infinite-dimensional Morse theory J. Differential Geom. 1989 207 221

[13] Floer, Andreas, Hofer, Helmut, Salamon, Dietmar Transversality in elliptic Morse theory for the symplectic action Duke Math. J. 1995 251 292

[14] Fukaya, Kenji Morse homotopy, 𝐴^{∞}-category, and Floer homologies 1993 1 102

[15] Gelfand, Sergei I., Manin, Yuri I. Methods of homological algebra 1996

[16] Goodman, Frederick M., Wenzl, Hans The Temperley-Lieb algebra at roots of unity Pacific J. Math. 1993 307 334

[17] Haefliger, André Plongements différentiables de variétés dans variétés Comment. Math. Helv. 1961 47 82

[18] Hofer, H. Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three Invent. Math. 1993 515 563

[19] Kontsevich, Maxim Homological algebra of mirror symmetry 1995 120 139

[20] Martin, Paul Potts models and related problems in statistical mechanics 1991

[21] Mcduff, Dusa, Salamon, Dietmar Introduction to symplectic topology 1995

[22] Moody, John Atwell The Burau representation of the braid group 𝐵_{𝑛} is unfaithful for large 𝑛 Bull. Amer. Math. Soc. (N.S.) 1991 379 384

[23] Oh, Yong-Geun Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I Comm. Pure Appl. Math. 1993 949 993

[24] Oh, Yong-Geun On the structure of pseudo-holomorphic discs with totally real boundary conditions J. Geom. Anal. 1997 305 327

[25] Poźniak, Marcin Floer homology, Novikov rings and clean intersections 1999 119 181

[26] Robbin, Joel, Salamon, Dietmar The Maslov index for paths Topology 1993 827 844

[27] Salamon, Dietmar Morse theory, the Conley index and Floer homology Bull. London Math. Soc. 1990 113 140

[28] Salamon, Dietmar, Zehnder, Eduard Morse theory for periodic solutions of Hamiltonian systems and the Maslov index Comm. Pure Appl. Math. 1992 1303 1360

[29] Seidel, Paul Graded Lagrangian submanifolds Bull. Soc. Math. France 2000 103 149

[30] Seidel, Paul Lagrangian two-spheres can be symplectically knotted J. Differential Geom. 1999 145 171

[31] Viterbo, Claude Intersection de sous-variétés lagrangiennes, fonctionnelles d’action et indice des systèmes hamiltoniens Bull. Soc. Math. France 1987 361 390

[32] Weinstein, Alan Lagrangian submanifolds and Hamiltonian systems Ann. of Math. (2) 1973 377 410

[33] Westbury, B. W. The representation theory of the Temperley-Lieb algebras Math. Z. 1995 539 565