International Congress of Mathematicians

Fukaya Categories and Deformations 353normalization contains some of the first p and none of the last q punctures. Thismeans that if we take only genus zero and q = 1 then no degenerations at all areallowed, and the resulting structure is that of a Batalin-Vilkovisky (BV) algebra[10]. For instance, let M = D(T*L) be a unit cotangent bundle of an oriented closedmanifold L. Viterbo [27] computed that SH*(M) = ff„_»(AL) is the homology ofthe free loop space, and a reasonable conjecture says that the BV structure agreeswith that of Chas-Sullivan [1].Returning to the specific situation of Assumption 1, and supposing that U hasbeen chosen in such a way that the Reeb flow on dM becomes periodic, one can usea Bott-Morse argument [19] to get a spectral sequence which converges to SH*(M).The starting term isEPi = i H9 ( M ) P=°> (1)1\Hi+ 3 P(dM)p 2 (andappealing to hard Lefschetz, which will be the only time that we use any algebraicgeometry) one hasdimSH 2 (M) < b 2 (M) + b 0 (dM) = b 2 (X). (2){'2. Fukaya categoriesM (taken as in Assumption 1) is an exact symplectic manifold, and there is awell-defined notion of exact Lagrangian submanifold in it. Such submanifolds L havethe property that there are no non-constant holomorphic maps u : (£, 9S) —t (M, L)for a compact Riemann surface S, hence a theory of "Gromov-Witten invariantswith Lagrangian boundary conditions" would be trivial in this case. To get somethinginteresting, one removes some boundary points from S, thus dividing theboundary into several components, and assigns different L to them. The part ofthis theory where S is a disk gives rise to the Fukaya A^-category J(M).The basic algebraic notion is as follows. An A^-category A (over some field,let's say Q) consists of a set of objects Ob A, and for any two objects a gradedQ-vector space of morphisms hom A (Xo,Xi), together with composition operationsp A : hom A (X 0 ,Xi) —y hom A (X 0 , Xi)[l],p 2 A : hom A (Xi,X 2 ) ® hom A (X 0 ,Xi) —y hom A (Xo,X 2 ),p? A : hom A (X 2 ,X 3 ) hom A (Xi,X 2 ) ® hom A (X 0 ,Xi) ->—>hom A (X 0 ,X 3 )[-l], ....These must satisfy a sequence of quadratic "associativity" equations, which ensurethat p} A is a differential, p A a morphism of chain complexes, and so on. Note that by