On Landau–Ginzburg systems, quivers and monodromy

Journal of Mathematical Physics, 2017

Let X be a toric Del-Pezzo surface and let Crit(W) ⊂ (C *) n be the solution scheme of the Landau-Ginzburg system of equations. Denote by X • the polar variety of X. Our aim in this work is to describe a map L : Crit(W) → F uk trop (X •) whose image under homological mirror symmetry corresponds to a full strongly exceptional collection of line bundles.

Cornell University - arXiv, 2016

For a toric Fano manifold X denote by Crit(X) ⊂ (C *) n the solution scheme of the Landau-Ginzburg system of equations of X. Examples of toric Fano manifolds with rk(P ic(X)) ≤ 3 which admit full strongly exceptional collections of line bundles were recently found by various authors. For these examples we construct a map E : Crit(X) → P ic(X) whose image E = {E(z)|z ∈ Crit(X)} is a full strongly exceptional collection satisfying the Maligned property. That is, under this map, the groups Hom(E(z), E(w)) for z, w ∈ Crit(X) are naturally related to the structure of the monodromy group acting on Crit(X).

Cornell University - arXiv, 2014

r i=1 O P s (a i)) be a Fano projective bundle over P s and denote by Crit(X) ⊂ (C *) n the solution scheme of the Landau-Ginzburg system of equations of X. We describe a map E : Crit(X) → P ic(X) whose image E = {E(z)|z ∈ Crit(X)} is the full strongly exceptional collection described by Costa and Miró-Roig in [15]. We further show that Hom(E(z), E(w)) for z, w ∈ Crit(X) can be described in terms of a monodromy group acting on Crit(X).

Journal of Algebra, 2013

We show that every Picard rank one smooth Fano threefold has a weak Landau-Ginzburg model coming from a toric degeneration. The fibers of these Landau-Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of Cartier divisors in weighted projective space has a very weak Landau-Ginzburg model coming from a toric degeneration.

2002

We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture which has close relations to toric mirror symmetry. Our conjecture, we call it Toric Residue Mirror Conjecture, claims that the generating functions of intersection numbers of divisors on a special sequence of simplicial toric varieties are power series expansions of some rational functions obtained as toric residues. We expect that this conjecture holds true for all Gorenstein toric Fano varieties associated with reflexive polytopes and give some evidences for that. The proposed conjecture suggests a simple method for computing Yukawa couplings for toric mirror Calabi-Yau hypersurfaces without solving systems of differential equations. We make several explicit computations for Calabi-Yau hypersurfaces in weighted projective spaces and in products of projective spaces.

2012

Nuclear Physics B, 1995

The moduli dependence of (2; 2) superstring compactications based on Calabi{ Yau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg orbifolds with c = 9 whose potential is a sum of A-type singularities. Here we consider the generalization to arbitrary quasi-homogeneous singularities at c = 9 . W e use mirror symmetry to derive the dependence of the models on the complexied K ahler moduli and check the expansions of some topological correlation functions against explicit genus zero and genus one instanton calculations. As an important application we give examples of how non-algebraic (\twisted") deformations can be mapped to algebraic ones, hence allowing us to study the full moduli space. We also study how moduli spaces can be nested in each other, thus enabling a (singular) transition from one theory to another. Following the recent w ork of Greene, Morrison and Strominger we show that this corresponds to black hole condensation in type II string theories compactied on Calabi-Yau manifolds.

2012

We show which of the smooth Fano threefolds admit degenerations to toric Fano threefolds with ordinary double points.

arXiv: Algebraic Geometry, 2019

We extend the definition of Noether-Leschetz components to quasi-smooth hypersurfaces in a projective simplicial toric variety $\mathbb P_{\Sigma}^{2k+1}$, and prove that asymptoticaly the components whose codimension is bounded from above by a suitable effective constant correspond to hypersurfaces containing a small degree $k$-dimensional subvariety. As a corollary we get an asymptotic characterization of the components with small codimension, generalizing the work of Otwinowska for $\mathbb P_{\Sigma}^{2k+1}=\mathbb P^{2k+1}$ and Green and Voisin for $\mathbb P_{\Sigma}^{2k+1}=\mathbb P^3$. Some tools that are developed in the paper are a generalization of Macaulay's theorem for Fano, irreducible normal varieties with rational singularities, satisfying a suitable additional condition, and an extension of the notion of Gorenstein ideal for normal varieties with finitely generated Cox ring.

arXiv (Cornell University), 2023

The cellular resolution of the diagonal given by Bayer-Popescu-Sturmfels for unimodular projective toric varieties yields a full, strong exceptional collection of line bundles on unimodular projective toric surfaces. The Hanlon-Hicks-Lazarev resolution of the diagonal yields a full, strong exceptional collection of line bundles for 16 of the 18 smooth toric Fano threefolds.