On Landau-Ginzburg systems, quivers and monodromy

JOURNAL OF GEOMETRY AND PHYSICS

卷:98

Article

Jerby, Yochay¹ 

[1] Inst Math, CH-2000 Neuchatel, Switzerland

关键词: Quantum cohomology;    Exceptional collections;    Mirror symmetry;    Landau-Ginzburg system;    Toric Fano manifolds;   

DOI  :  10.1016/j.geomphys.2015.09.006

来源: Elsevier

【 摘 要 】

Let X be a toric Fano manifold and denote by Crit(f(x)) subset of (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(f(x)) -> Pic(X) such that epsilon(L)(X) := L(Crit(f(x))) subset of 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) \ R-x, f(x)) -> 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 of 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 epsilon(L)(X). We refer to this property as the M-aligned property of the maps L : Crit(fx) -> 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. (C) 2015 Elsevier B.V. All rights reserved.

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