On Landau–Ginzburg systems, quivers and monodromy

Abstract

Let X be a toric Fano manifold and denote by C r i t ( f X ) ⊂ ( 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 : C r i t ( f X ) → P i c ( X ) such that E L ( X ) : = L ( C r i t ( f X ) ) ⊂ P i c ( X ) is a full strongly exceptional collection of line bundles. We observe the existence of a natural monodromy map M : π 1 ( L ( X ) ∖ R X , f X ) → A u t ( C r i t ( 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 ⊂ L ( X ) is the space of all elements whose corresponding solution scheme is reduced. We show that monodromies of C r i t ( f X ) admit non-trivial relations to quiver representations of the exceptional collection E L ( X ) . We refer to this property as the M -aligned property of the maps L : C r i t ( f X ) → P i c ( X ) . We discuss possible applications of the existence of such M -aligned exceptional maps to various aspects of mirror symmetry of toric Fano manifolds.