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Autoequivalences vs. symplectomorphisms for Fukaya categories

Determine whether, for a symplectic manifold W, the group of autoequivalences (autofunctors) of its Fukaya category is isomorphic to the stabilized group of symplectic automorphisms of W. Establish this correspondence in full generality, clarifying the relationship between categorical automorphisms of the Fukaya category and geometric symplectomorphisms after stabilization.

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Background

The paper studies computable homotopy-theoretic constructions for dg categories and applies them to wrapped Fukaya categories of Weinstein manifolds, illustrating how symplectomorphisms induce functors between Fukaya categories via homotopy colimit presentations.

Within this context, a longstanding conjectural link posits that the group of autoequivalences of a Fukaya category mirrors the stabilized symplectomorphism group of the underlying symplectic manifold. Establishing this correspondence would imply that induced functors fully capture the geometric information of symplectic automorphisms, motivating the explicit computation of such induced functors as in the paper's examples.

References

The first motivation stems from a conjecture written in , which questions whether the group of autofunctors of the Fukaya category coincides with the (stabilized) group of symplectic automorphisms. If this conjecture holds true, the induced functor would encapsulate the same information as the original symplectic automorphism.

— A Computational Approach to the Homotopy Theory of DG categories (2405.03258 - Karabas et al., 6 May 2024) in Section 7 (Wrapped Fukaya category of T^*S^n and the reflection functor), introductory motivation paragraphs