International Congress of Mathematicians

Fukaya Categories and Deformations 357Lagrangian submanifoldsPeriodic orbit of theHamiltonianFigure 2:5. Deformations of categoriesThe following general definition, due to Kontsevich, satisfies the need for adeformation theory of categories which should be applicable to a wide range of situations:for instance, a deformation of a complex manifold should induce a deformationof the associated differential graded category of complexes of holomorphicvector bundles. By thinking about this example, one quickly realizes that sucha notion of deformation must include a change in the set of objects itself. TheAQO-formalism, slightly extended in an entirely natural way, fits that requirementperfectly. The relevance to symplectic topology is less immediately obvious, but itplays a central role in Fukaya, Oh, Ohta and Ono's work on "obstructions" in Floercohomology [9] (a good expository account from their point of view is [5]).For concreteness we consider only A^-deformations with one formal parameter,that is to say over Q[[t]]. Such a deformation £ is given by a set Ob £ of objects,and for any two objects a space homcfiXo, Xi) of morphisms which is a free gradedQ[[r]]-module, together with composition operations as before but now including a0-ary one: this consists of a so-called "obstruction cocycle"p\ £ hom\(X,X) (6)for every object X, and it must be of order t (no constant term). There is a sequenceof associativity equations, extending those of an A^-category by terms involvingp\. Clearly, if one sets t = 0 (by tensoring with Q over Q[[r]]), p% vanishes and theoutcome is an ordinary A^-category over Q. This is called the special fibre anddenoted by £ sp . One says that £ is a deformation of £ sp .A slightly more involved construction associates to £ two other A^-categories,the global section category £. g i and the generic fibre £. gen , which are defined overQ[[t]] and over the Laurent series ring Q[t _1 ][[t]], respectively. One first enlarges £to a bigger A^-deformation £ c by coupling the existing objects with formal connections(the terminology comes from the application to complexes of vector bundles).Objects of £ c are pairs (X,a) consisting of X £ Ob £ and an a e hom\(X, X)