On Landau-Ginzburg systems, Quivers and Monodromy

Let $X$ be a toric Fano manifold and denote by $Crit(f_X) \subset (\mathbb{C}^{\ast})^n$ the solution scheme of the corresponding Landau-Ginzburg system of equations. For toric Del-Pezzo surfaces and various toric Fano threefolds we define a map $L : Crit(f_X) \rightarrow Pic(X)$ such that $\mathcal{E}_L(X) : = L(Crit(f_X)) \subset Pic(X)$ is a full strongly exceptional collection of line bundles. We observe the existence of a natural monodromy map $$M : \pi_1(L(X) \setminus R_X,f_X) \rightarrow Aut(Crit(f_X))$$ where $L(X)$ is the space of all Laurent polynomials whose Newton polytope is equal to the Newton polytope of $f_X$, the Landau-Ginzburg potential of $X$, and $R_X \subset L(X)$ is the space of all elements whose corresponding solution scheme is reduced. We show that monodromies of $Crit(f_X)$ admit non-trivial relations to quiver representations of the exceptional collection $\mathcal{E}_L(X)$. We refer to this property as the $M$-aligned property of the maps $L: Crit(f_X) \rightarrow Pic(X)$. We discuss possible applications of the existence of such $M$-aligned exceptional maps to various aspects of mirror symmetry of toric Fano manifolds.