International Congress of Mathematicians

Fukaya Categories and Deformations 3597(X), because it contains only Lagrangian submanifolds which lie in M. However,that difference may disappear if one passes to derived categories:Conjecture 5. In the situation of Assumption 2, there is a canonical equivalenceof triangulated categoriesD*(Ï(M C X) gen ® Q[t -i] [[t] ] A t ) - D*(f(X)).In comparison with the previous conjecture, Assumption 2 is far more importanthere. The idea is that there should be an analogue of Theorem 3 for D*(9r(X)),saying that this category is split-generated by vanishing cycles, hence by objectswhich are also present in J(M c X).To pull together the various speculations, suppose that Y = CP" +1 for somen > 3; X c Y is a hypersurface of degree n + 2; and D c X is the intersectionof two such hypersurfaces. Then D 7 '('J(Mj) is split-generated by finitelymanyobjects, hence Tw 7 '('J(Mj) is at least in principle accessible to computation.Conjecture 4 together with (2), (4) tells us that HH 2 (7(M),'3 : (Mj) =ëHH 2 (Tw 7 '('J(Mj),Tw 7 '('J(Mjj) is at most one-dimensional, so an A,»-deformationof Tw 7 '('J(Mj) is unique up to a change of the parameter t. From this deformation,Conjecture 5 would enable one to find D*(9r(X)), again with the indeterminacy inthe parameter (fixing this is somewhat like computing the mirror map).Acknowledgements. Obviously, the ideas outlined here owe greatly to Fukaya andKontsevich. The author is equally indebted to Auroux, Donaldson, Getzler, Joyce,Khovanov, Smith, and Thomas (an incomplete list), all of whom have influenced histhinking considerably. The preparation of this talk at the Institute for AdvancedStudy was supported by NSF grant DMS-9729992.References[1] M. Chas and D. Sullivan, String topology, Preprint math.GT/9911159.[2] K. Cieliebak, A. Floer, and H. Hofer, Symplectic homology II: a general construction,Math. Z. 218 (1995), 103^122.[3] Ya. Eliashberg, A. Gi ventai, and H. Hofer, Introduction to symplectic fieldtheory, Geom. Funct. Anal. Special Volume, Part II (2000), 560^673.[4] K. Fukaya, Asymptotic analysis, multivalued Morse theory, and mirror symmetry,Preprint 2002.[5] , Deformation theory, homological algebra, and mirror symmetry,Preprint, December 2001.[6] , Floer homology and mirror symmetry II, Preprint 2001.[7] , Morse homotopy, .4 œ -categories, and Floer homologies, Proceedings ofGARC workshop on Geometry and Topology (H. J. Kim, ed.), Seoul NationalUniversity, 1993.[8] K. Fukaya and Y.-G. Oh, Zero-loop open strings in the cotangent bundle andMorse homotopy, Asian J. Math. 1 (1998), 96^180.[9] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian intersection Floertheory - anomaly and obstruction, Preprint, 2000.