The Fukaya category, exotic forms and exotic autoequivalences
A symplectic manifold is a smooth manifold M together with a choice of a closed non-degenerate two-form. Recent years have seen the importance of associating an A∞-category to M, called its Fukaya category, in helping to understand symplectic properties of M and its Lagrangian submanifolds. One of the principles of this construction is that automorphisms of the symplectic manifold should induce autoequivalences of the derived Fukaya category, although precisely what autoequivalences are thus obtained has been established in very few cases. Given a Lagrangian V ≅ CPn in a symplectic manifold (M,ω), there is an associated symplectomorphism ∅v of M. In Part I, we defi ne the notion of a CPn-object in an A∞-category A, and use this to construct algebraically an A∞- functor Φv , which we prove induces an autoequivalence of the derived category DA. We conjecture that Φv corresponds to the action of ∅v and prove this in the lowest dimension n = 1. We also give examples of symplectic manifolds for which this twist can be defi ned algebraically, but corresponds to no geometric automorphism of the manifold itself: we call such autoequivalences exotic. Computations in Fukaya categories have also been useful in distinguishing certain symplectic forms on exact symplectic manifolds from the "standard" forms. In Part II, we investigate the uniqueness of so-called exotic structures on certain exact symplectic manifolds by looking at how their symplectic properties change under small nonexact deformations of the symplectic form. This allows us to distinguish between two exotic symplectic forms on T*S3∪2-handle, even though the standard symplectic invariants such as their Fukaya category and their symplectic cohomology vanish. We also exhibit, for any n, an exact symplectic manifold with n distinct, exotic symplectic structures, which again cannot be distinguished by symplectic cohomology or by the Fukaya category.