On the mirrors of low-degree del Pezzo surfaces

TL;DR

This paper compares different mirror constructions for low-degree del Pezzo surfaces, extracting Lefschetz fibrations and relating categorical and geometric perspectives to deepen understanding of their mirror symmetry.

Contribution

It introduces a method to relate various mirror constructions via Lefschetz fibrations and mutations, connecting categorical and geometric approaches for del Pezzo surfaces.

Findings

Lefschetz fibrations define categorical mirrors.

Explicit mutations relate different exceptional collections.

Fukaya-Seidel categories are shown to be equivalent.

Abstract

We compare different constructions of mirrors of del Pezzo surfaces, focusing on degree $d \leq 3$. In particular, we extract Lefschetz fibrations, with associated exceptional collections, from the mirrors obtained via the Hori-Vafa and Fanosearch program constructions, which we relate to one another. We show with geometric methods that the Lefschetz fibrations define categorical mirrors. With a more explicit approach, we give a sequence of (numerical) mutations relating the exceptional collections considered by Auroux, Katzarkov, and Orlov with those arising in this paper. This uses the theory of surface-like pseudolattices, and extends some of the string junction results of Grassi, Halverson and Shaneson. Our argument lifts directly to an equivalence of certain Fukaya-Seidel categories arising from our fibrations and those of Auroux, Katzarkov, and Orlov.