On Landau–Ginzburg systems, quivers and monodromy

Let XX be a toric Fano manifold and denote by Crit(fX)⊂(C∗)nCrit(fX)⊂(C∗)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(fX)→Pic(X)L:Crit(fX)→Pic(X) such that EL(X):=L(Crit(fX))⊂Pic(X)EL(X):=L(Crit(fX))⊂Pic(X) is a full strongly exceptional collection of line bundles. We observe the existence of a natural monodromy map M:π1(L(X)∖RX,fX)→Aut(Crit(fX))M:π1(L(X)∖RX,fX)→Aut(Crit(fX)) where L(X)L(X) is the space of all Laurent polynomials whose Newton polytope is equal to the Newton polytope of fXfX, the Landau–Ginzburg potential of XX, and RX⊂L(X)RX⊂L(X) is the space of all elements whose corresponding solution scheme is reduced. We show that monodromies of Crit(fX)Crit(fX) admit non-trivial relations to quiver representations of the exceptional collection EL(X)EL(X). We refer to this property as the MM-aligned property of the maps L:Crit(fX)→Pic(X)L:Crit(fX)→Pic(X). We discuss possible applications of the existence of such MM-aligned exceptional maps to various aspects of mirror symmetry of toric Fano manifolds.