Geodesic
Fukaya Categories as Categorical Morse Homology
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The Fukaya category of a Weinstein manifold is an intricate symplectic invariant of high interest in mirror symmetry and geometric representation theory. This paper informally sketches how, in analogy with Morse homology, the Fukaya category might result from gluing together Fukaya categories of Weinstein cells. This can be formalized by a recollement pattern for Lagrangian branes parallel to that for constructible sheaves. Assuming this structure, we exhibit the Fukaya category as the global sections of a sheaf on the conic topology of the Weinstein manifold. This can be viewed as a symplectic analogue of the well-known algebraic and topological theories of (micro)localization.
Keywords: Fukaya category; microlocalization. 
@article{SIGMA_2014_10_a17,
     author = {David Nadler},
     title = {Fukaya {Categories} as {Categorical} {Morse} {Homology}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2014},
     volume = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a17/}
}
TY  - JOUR
AU  - David Nadler
TI  - Fukaya Categories as Categorical Morse Homology
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2014
VL  - 10
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a17/
LA  - en
ID  - SIGMA_2014_10_a17
ER  - 
%0 Journal Article
%A David Nadler
%T Fukaya Categories as Categorical Morse Homology
%J Symmetry, integrability and geometry: methods and applications
%D 2014
%V 10
%U http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a17/
%G en
%F SIGMA_2014_10_a17
David Nadler. Fukaya Categories as Categorical Morse Homology. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a17/

[1] Abouzaid M., “A topological model for the {F}ukaya categories of plumbings”, J. Differential Geom., 87 (2011), 1–80, arXiv: 0904.1474 | MR | Zbl

[2] Abouzaid M., “On the wrapped {F}ukaya category and based loops”, J. Symplectic Geom., 10 (2012), 27–79, arXiv: 0904.1474 | DOI | MR | Zbl

[3] Abouzaid M., Auroux D., Efimov A. I., Katzarkov L., Orlov D., “Homological mirror symmetry for punctured spheres”, J. Amer. Math. Soc., 26 (2013), 1051–1083, arXiv: 1103.4322 | DOI | MR | Zbl

[4] Abouzaid M., Seidel P., “An open string analogue of {V}iterbo functoriality”, Geom. Topol., 14 (2010), 627–718, arXiv: 0712.3177 | DOI | MR | Zbl

[5] Audin M., Lalonde F., Polterovich L., “Symplectic rigidity: {L}agrangian submanifolds”, Holomorphic Curves in Symplectic Geometry, Progr. Math., 117, Birkhäuser, Basel, 1994, 271–321 | MR

[6] Auroux D., “Fukaya categories and bordered {H}eegaard–{F}loer homology”, Proceedings of the {I}nternational {C}ongress of {M}athematicians, v. II, Hindustan Book Agency, New Delhi, 2010, 917–941, arXiv: 1003.2962 | MR

[7] Auroux D., “Fukaya categories of symmetric products and bordered {H}eegaard–{F}loer homology”, J. Gökova Geom. Topol. GGT, 4 (2010), 1–54, arXiv: 1001.4323 | MR | Zbl

[8] Beilinson A., Bernstein J., “A proof of {J}antzen conjectures”, I. {M}. {G}el'fand {S}eminar, Adv. Soviet Math., 16, Amer. Math. Soc., Providence, RI, 1993, 1–50 | MR | Zbl

[9] Beilinson A. A., “Coherent sheaves on {$P^{n}$} and problems in linear algebra”, Funct. Anal. Appl., 12 (1978), 214–216 | DOI | MR

[10] Ben-Zvi D., Francis J., Nadler D., “Integral transforms and {D}rinfeld centers in derived algebraic geometry”, J. Amer. Math. Soc., 23 (2010), 909–966, arXiv: 0805.0157 | DOI | MR | Zbl

[11] Bierstone E., Milman P. D., “Semianalytic and subanalytic sets”, Inst. Hautes Études Sci. Publ. Math., 1988, 5–42 | DOI | MR | Zbl

[12] Biran P., “Lagrangian barriers and symplectic embeddings”, Geom. Funct. Anal., 11 (2001), 407–464 | DOI | MR | Zbl

[13] Biran P., “Lagrangian non-intersections”, Geom. Funct. Anal., 16 (2006), 279–326, arXiv: math.SG/0412110 | DOI | MR | Zbl

[14] Bourgeois F., Ekholm T., Eliashberg Y., “Symplectic homology product via {L}egendrian surgery”, Proc. Natl. Acad. Sci. USA, 108 (2011), 8114–8121, arXiv: 1012.4218 | DOI | MR | Zbl

[15] Bourgeois F., Ekholm T., Eliashberg Y., “Effect of {L}egendrian surgery”, Geom. Topol., 16 (2012), 301–389, arXiv: 0911.0026 | DOI | MR | Zbl

[16] Braden T., Licata A., Proudfoot N., Webster B., “Hypertoric category {${\mathcal O}$}”, Adv. Math., 231 (2012), 1487–1545, arXiv: 1010.2001 | DOI | MR | Zbl

[17] Cieliebak K., “Handle attaching in symplectic homology and the chord conjecture”, J. Eur. Math. Soc., 4 (2002), 115–142 | DOI | MR | Zbl

[18] Cieliebak K., Eliashberg Y., Symplectic geometry of Stein manifolds, unpublished, 2006

[19] Eliashberg Y., “Symplectic geometry of plurisubharmonic functions”, Gauge Theory and Symplectic Geometry ({M}ontreal, {PQ}, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 488, Kluwer Acad. Publ., Dordrecht, 1997, 49–67 | MR | Zbl

[20] Eliashberg Y., Gromov M., “Convex symplectic manifolds”, Several Complex Variables and Complex Geometry ({S}anta {C}ruz, {CA}, 1989), v. 2, Proc. Sympos. Pure Math., 52, Amer. Math. Soc., Providence, RI, 1991, 135–162 | DOI | MR

[21] Fukaya K., Oh Y.-G., “Zero-loop open strings in the cotangent bundle and {M}orse homotopy”, Asian J. Math., 1 (1997), 96–180 | MR | Zbl

[22] Fukaya K., Oh Y.-G., Ohta H., Ono K., Lagrangian intersection Floer theory — anomaly and obstruction, {K}yoto Preprint, Math 00-17, 2000

[23] Gordon I., Stafford J. T., “Rational {C}herednik algebras and {H}ilbert schemes”, Adv. Math., 198 (2005), 222–274, arXiv: math.RA/0407516 | DOI | MR | Zbl

[24] Gordon I., Stafford J. T., “Rational {C}herednik algebras and {H}ilbert schemes. {II}: {R}epresentations and sheaves”, Duke Math. J., 132 (2006), 73–135, arXiv: math.RT/0410293 | DOI | MR | Zbl

[25] Gotay M. J., “On coisotropic imbeddings of presymplectic manifolds”, Proc. Amer. Math. Soc., 84 (1982), 111–114 | DOI | MR | Zbl

[26] Kashiwara M., Rouquier R., “Microlocalization of rational {C}herednik algebras”, Duke Math. J., 144 (2008), 525–573, arXiv: 0705.1245 | DOI | MR | Zbl

[27] Kashiwara M., Schapira P., Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften, 292, Springer-Verlag, Berlin, 1994 | MR

[28] Khovanov M., Seidel P., “Quivers, {F}loer cohomology, and braid group actions”, J. Amer. Math. Soc., 15 (2002), 203–271, arXiv: math.QA/0006056 | DOI | MR | Zbl

[29] Kontsevich M., Symplectic geometry of homological algebra, 2009 http://www.ihes.fr/m̃axim/TEXTS/Symplectic_AT2009.pdf

[30] Kostant B., “On {W}hittaker vectors and representation theory”, Invent. Math., 48 (1978), 101–184 | DOI | MR | Zbl

[31] Loday J. L., “The diagonal of the {S}tasheff polytope”, Higher Structures in Geometry and Physics, Progr. Math., 287, Birkhäuser/Springer, New York, 2011, 269–292, arXiv: 0710.0572 | DOI | MR | Zbl

[32] Losev I., “Finite {$W$}-algebras”, Proceedings of the {I}nternational {C}ongress of {M}athematicians, v. III, Hindustan Book Agency, New Delhi, 2010, 1281–1307, arXiv: 1003.5811 | MR

[33] Losev I., “Quantized symplectic actions and {$W$}-algebras”, J. Amer. Math. Soc., 23 (2010), 35–59, arXiv: 0707.3108 | DOI | MR | Zbl

[34] Lurie J., Higher algebra, http://www.math.harvard.edu/l̃urie/

[35] Lurie J., Higher topos theory, Annals of Mathematics Studies, 170, Princeton University Press, Princeton, NJ, 2009, arXiv: math.CT/0608040 | MR | Zbl

[36] Lurie J., “On the classification of topological field theories”, Current Developments in Mathematics 2008, Int. Press, Somerville, MA, 2009, 129–280, arXiv: 0905.0465 | MR | Zbl

[37] Manolescu C., “Nilpotent slices, {H}ilbert schemes, and the {J}ones polynomial”, Duke Math. J., 132 (2006), 311–369, arXiv: math.SG/0411015 | DOI | MR | Zbl

[38] Markl M., Shnider S., “Associahedra, cellular {$W$}-construction and products of {$A_\infty$}-algebras”, Trans. Amer. Math. Soc., 358 (2006), 2353–2372, arXiv: math.AT/0312277 | DOI | MR | Zbl

[39] Ma'u S., Gluing pseudoholomorphic quilted disks, arXiv: 0909.3339

[40] McGerty K., Nevins T., Derived equivalence for quantum symplectic resolutions, arXiv: 1108.6267 | MR

[41] Nadler D., “Microlocal branes are constructible sheaves”, Selecta Math. (N.S.), 15 (2009), 563–619, arXiv: math.SG/0612399 | DOI | MR | Zbl

[42] Nadler D., “Springer theory via the {H}itchin fibration”, Compos. Math., 147 (2011), 1635–1670, arXiv: 0806.4566 | DOI | MR | Zbl

[43] Nadler D., Zaslow E., “Constructible sheaves and the {F}ukaya category”, J. Amer. Math. Soc., 22 (2009), 233–286, arXiv: math.SG/0604379 | DOI | MR | Zbl

[44] Oh Y.-G., Unwrapped continuation invariance in Lagrangian Floer theory: energy and $C^0$ estimates, arXiv: 0910.1131

[45] Perutz T., “Lagrangian matching invariants for fibred four-manifolds, {I}”, Geom. Topol., 11 (2007), 759–828, arXiv: math.SG/0606061 | DOI | MR | Zbl

[46] Perutz T., “Lagrangian matching invariants for fibred four-manifolds, {II}”, Geom. Topol., 12 (2008), 1461–1542, arXiv: math.SG/0606062 | DOI | MR | Zbl

[47] Perutz T., A symplectic Gysin sequence, arXiv: 0807.1863

[48] Premet A., “Special transverse slices and their enveloping algebras”, Adv. Math., 170 (2002), 1–55 | DOI | MR | Zbl

[49] Saneblidze S., Umble R., A diagonal on the associahedra, arXiv: math.AT/0011065

[50] Saneblidze S., Umble R., “Diagonals on the permutahedra, multiplihedra and associahedra”, Homology Homotopy Appl., 6 (2004), 363–411, arXiv: math.AT/0209109 | DOI | MR | Zbl

[51] Seidel P., “More about vanishing cycles and mutation”, Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci. Publ., River Edge, NJ, 2001, 429–465, arXiv: math.SG/0010032 | DOI | MR | Zbl

[52] Seidel P., “Vanishing cycles and mutation”, European {C}ongress of {M}athematics (Barcelona, 2000), v. II, Progr. Math., 202, Birkhäuser, Basel, 2001, 65–85, arXiv: math.SG/0007115 | MR | Zbl

[53] Seidel P., “A long exact sequence for symplectic {F}loer cohomology”, Topology, 42 (2003), 1003–1063, arXiv: math.SG/0105186 | DOI | MR | Zbl

[54] Seidel P., “{$A_\infty$}-subalgebras and natural transformations”, Homology, Homotopy Appl., 10 (2008), 83–114, arXiv: math.KT/0701778 | DOI | MR | Zbl

[55] Seidel P., Fukaya categories and {P}icard–{L}efschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008, arXiv: math.SG/0206155 | DOI | MR | Zbl

[56] Seidel P., “Symplectic homology as {H}ochschild homology”, Algebraic Geometry (Seattle, 2005), v. 1, Proc. Sympos. Pure Math., 80, Amer. Math. Soc., Providence, RI, 2009, 415–434, arXiv: math.SG/0609037 | DOI | MR | Zbl

[57] Seidel P., Cotangent bundles and their relatives, morse Lectures, , Princeton University, 2010 http://www-math.mit.edu/s̃eidel/talks/morse-lectures-1.pdf

[58] Seidel P., “Homological mirror symmetry for the genus two curve”, J. Algebraic Geom., 20 (2011), 727–769, arXiv: 0812.1171 | DOI | MR | Zbl

[59] Seidel P., “Some speculations on pairs-of-pants decompositions and {F}ukaya categories”, Surveys in Differential Geometry, 17, Int. Press, Boston, MA, 2012, 411–425, arXiv: 1004.0906 | DOI | MR

[60] Seidel P., Smith I., “A link invariant from the symplectic geometry of nilpotent slices”, Duke Math. J., 134 (2006), 453–514, arXiv: math.SG/0405089 | DOI | MR | Zbl

[61] Sheridan N., “On the homological mirror symmetry conjecture for pairs of pants”, J. Differential Geom., 89 (2011), 271–367, arXiv: 1012.3238 | MR | Zbl

[62] Sibilla N., Treumann D., Zaslow E., Ribbon graphs and mirror symmetry, I, arXiv: 1103.2462

[63] Sikorav J. C., “Some properties of holomorphic curves in almost complex manifolds”, Holomorphic curves in symplectic geometry, Progr. Math., 117, Birkhäuser, Basel, 1994, 165–189 | MR

[64] Stasheff J. D., “Homotopy associativity of {$H$}-spaces, {I}”, Trans. Amer. Math. Soc., 108 (1963), 275–292 | DOI | MR | Zbl

[65] Stasheff J. D., “Homotopy associativity of {$H$}-spaces, {II}”, Trans. Amer. Math. Soc., 108 (1963), 293–312 | DOI | MR | Zbl

[66] van den Dries L., Miller C., “Geometric categories and o-minimal structures”, Duke Math. J., 84 (1996), 497–540 | DOI | MR | Zbl

[67] Wehrheim K., Woodward C. T. http://math.berkeley.edu/k̃atrin/

[68] Wehrheim K., Woodward C., Pseudoholomorphic quilts, arXiv: 0905.1369

[69] Wehrheim K., Woodward C. T., “Functoriality for {L}agrangian correspondences in {F}loer theory”, Quantum Topol., 1 (2010), 129–170, arXiv: 0708.2851 | DOI | MR | Zbl

[70] Wehrheim K., Woodward C. T., “Quilted {F}loer cohomology”, Geom. Topol., 14 (2010), 833–902, arXiv: 0905.1370 | DOI | MR | Zbl

[71] Wehrheim K., Woodward C. T., “Floer cohomology and geometric composition of {L}agrangian correspondences”, Adv. Math., 230 (2012), 177–228, arXiv: 0905.1368 | DOI | MR | Zbl

[72] Wehrheim K., Woodward C. T., “Quilted {F}loer trajectories with constant components: corrigendum to the article “{Q}uilted {F}loer cohomology””, Geom. Topol., 16 (2012), 127–154, arXiv: 1101.3770 | DOI | MR | Zbl

[73] Weinstein A., Lectures on symplectic manifolds, Regional Conference Series in Mathematics, 29, Amer. Math. Soc., Providence, R.I., 1977 | MR | Zbl

[74] Weinstein A., “Neighborhood classification of isotropic embeddings”, J. Differential Geom., 16 (1981), 125–128 | MR | Zbl

[75] Weinstein A., “Contact surgery and symplectic handlebodies”, Hokkaido Math. J., 20 (1991), 241–251 | DOI | MR | Zbl