On Landau-Ginzburg Systems and $\mathcal{D}^b(X)$ of projective bundles

TL;DR

This paper explores the relationship between Landau-Ginzburg systems and the derived category of projective bundles, establishing a correspondence between critical points and exceptional collections, and analyzing monodromy actions.

Contribution

It introduces a map from Landau-Ginzburg critical points to the Picard group that recovers known exceptional collections on projective bundles.

Findings

Constructs a map from Crit(X) to Pic(X) linking critical points to exceptional objects.

Shows that Hom spaces between objects can be described via monodromy group actions.

Provides a geometric interpretation of the derived category structure in terms of Landau-Ginzburg models.

Abstract

Let $X=\mathbb{P}(\mathcal{O}_{\mathbb{P}^s} \oplus \bigoplus_{i=1}^r \mathcal{O}_{\mathbb{P}^s}(a_i))$ be a Fano projective bundle over $\mathbb{P}^s$ and denote by $Crit(X) \subset (\mathbb{C}^{\ast})^n$ the solution scheme of the Landau-Ginzburg system of equations of $X$. We describe a map $E : Crit(X) \rightarrow Pic(X)$ whose image $\mathcal{E}= \{E(z) | z \in Crit(X) \}$ is the full strongly exceptional collection on $X$ found by Costa and Mir$\acute{\textrm{o}}$-Roig. We further show that $Hom(E(z),E(w))$ for $z,w \in Crit(X)$ can be described in terms of a monodromy group acting on $Crit(X)$.