  We consider the action of symplectic monodromy on chain-level enhancements of
quantum cohomology. First, we construct a family version of
$A_\infty$-structure on quantum cohomology (this should morally correspond to
Hochschild cohomology of a "family of Fukaya categories over the circle").
Following Kaledin, we look at the obstruction class of this structure, and
argue that it can be related to a quantum version of Massey products on the one
hand (which, in nice cases, can be related to actual counts of rational curves)
and to the classical Andreadakis-Johnson theory of the Torelli group on the
other hand.
  In the second part of the paper, we go hunting for exotic symplectomorphism:
these are elements of infinite order in the kernel $$\mathcal{K}(M,\omega) :=
\pi_0 \mathrm{Symp}(M,\omega) \to \pi_0 \mathrm{Diff}^+(M,\omega)$$ of the
forgetful map from the symplectic mapping class group to the ordinary MCG. We
demonstrate how we can apply the theory above to prove the existence of such
elements $\psi_Y$ for certain a Fano 3-fold obtained by blowing-up
$\mathbb{P}^3$ at a genus 4 curve. Unlike the four-dimensional case, no power
of a Dehn twist around Lagrangian 3-spheres can be exotic (because they have
infinite order in smooth MCG).
  In the final part of the paper, the classical connection between our Fano
varieties and cubic 3-folds allows us to prove the existence of a new
phenomena: "exotic relations" in the subgroup generated by all Dehn twists.
Namely, it turns out we can factor some power of $[\psi_Y]$ in $\pi_0
\mathrm{Symp}(Y,\omega)$ into 3-dimensional Dehn twists. So the isotopy class
of the product in the ordinary MCG is torsion, but of infinite order in the
symplectic MCG.
