The quantum Johnson homomorphism and symplectomorphism of 3-folds

Abstract: 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.