This talk is about the symplectic isotopy problem: investigating the kernel of the forgetful map 

$$\pi_0 Symp(M,\omega) \to \pi_0 Diff^+(M)$$ 

when $(M,\omega)$ is a symplectic manifold. While many results are known for some time for $dim=4$, very little has been discovered in higher dimensions. 

Fukaya categories are a great way to organize the entire Floer theory of a manifold into one algebraic gadget, but they are not the ideal tool for every mission. We use a parametrized version of quantum Massey products and some $A_\infty$-deformation theory to introduce a quantum analogue of the Johnson homomorphism (familiar to low-dimensional topologist from the study of the Torelli group). 

This invariant attaches a ''characteristic class" in Hochschild cohomology to every suitable automorphism. Moreover, in good cases, this class be evaluated by studying the interaction of rational homotopy with Moduli spaces of holomorphic spheres. 

As a sample application, we consider $X = BL_C \mathbb{P}^3$, the blowup of projective space at a genus four curve. Using ideas from the birational geometry, we construct a symplectomorphism $\phi : X \to X$ with an explicit factorization as a product of six-dimensional Dehn twists. Even though each of the Dehn twists has infinite order in the symplectic mapping class group, we prove that $\phi$ is exotic.