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Given a symplectic manifold $(M,\omega)$, there is a natural map $$ Symp(M,\omega) \to Auteq(D^\pi Fuk(M,\omega))$$ which sends a symplectic automorphism to the $A_\infty$-functor it induces on the Fukaya category (I'm being delibrately vague about which flavor of Fukaya category are we talking about here).

  1. Is there any known case of a nontrivial symplectomorphism whose image is the identity functor?
  2. Does it matter if we replace $D^\pi Fuk(M,\omega)$ with $Fuk(M,\omega)$?
  3. What about if we replace the left-hand side with $\pi_0 Symp(M,\omega)$ (symplectomorphism up to Hamiltonian isotopy) and the right-hand side with $A_\infty$-functors up to $A_\infty$-homotopy?
  4. Is there any (physics?) reason to believe that the representation above should actually be faithful for certain manifolds?
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  • $\begingroup$ Since $D^\pi\mathcal F$ is defined functorially in $\mathcal F$, it clearly won't see anything more than $\mathcal F$. $\endgroup$
    – John Pardon
    Commented Mar 3, 2017 at 4:37
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    $\begingroup$ Yes, I agree - but it might see less... $\endgroup$
    – Nati
    Commented Mar 3, 2017 at 14:29
  • $\begingroup$ I'll change the order of 1. and 2. to make it clearer. $\endgroup$
    – Nati
    Commented Mar 3, 2017 at 16:19
  • $\begingroup$ What happens when you take HH? $\endgroup$
    – AHusain
    Commented Mar 9, 2017 at 2:24
  • $\begingroup$ @AHusain: If you mean Hochschild cohomology, it is commonly believed that you get the action of $\pi_0 Symp$ on the quantum cohomology $QH^\bullet(M,\omega)$ (endowed with the structure of quantum Massey products and considered as an $A_\infty$-algebra). $\endgroup$
    – Nati
    Commented Mar 19, 2017 at 5:11

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