There is a well-known theorem stating that there is a bijection between diffeomorphism classes of Lefschetz fibrations over $S^2$ whose general fiber is a closed orientable surface $\Sigma\_g$ of genus $g\geq 2$ and factorizations of identity in the mapping class group $\Gamma(\Sigma\_g)$ up to Hurwitz moves and global conjugation. It is explained [here](http://www.math.harvard.edu/theses/senior/parker/parker.pdf), for example. 


Is there an analogue of this theorem for symplectic Lefschetz fibrations in higher dimensions? In other words, are symplectomorphism classes of symplectic Lefschetz fibrations in bijective correspondence with factorizations of identity in the symplectic mapping class group of the general fiber?



Symplectic Lefschetz fibrations in terms of factorization in symplectic mapping class group
Up to homotopy, giving a Lefschetz fibration over $S^2=\mathbb C\cup\{\infty\}$ with singular fibers over $\{e^{2\pi ik/N}\}\_{0\leq k<N}$ is the same as giving the data of the fiber $F$ over $0\in\mathbb C$, the vanishing cycles $\{L\_i\}\_{0\leq i<N}$ (as marked Lagrangian spheres in $F$), and a path in $\operatorname{Symp}(F)$ from the identity to the product of Dehn twists about the $L\_i$.


Note the importance of remembering the path rather than just equality in $\pi\_0$: symplectic fibrations over $S^2$ with fiber $F$ over the basepoint, with no critical points, are classified by elements of $\pi\_1\operatorname{Symp}(F)$ via the clutching construction. This does not contradict the statement you give for $F=\Sigma\_g$ since the components of $\operatorname{Symp}(\Sigma\_g)$ are contractible.


The discussion of Hurwitz moves is the same as in the case of surfaces you mention. There is some discussion in sections 15-16 Seidel's book which may be helpful.


