|Title: ||The Fukaya category, exotic forms and exotic autoequivalences|
|Authors: ||Harris, Richard|
|Supervisors: ||Smith, Ivan|
|Issue Date: ||10-Apr-2012|
|Abstract: ||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.|
|Appears in Collections:||Theses - DPMMS|
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