Quantum differential systems and some applications to mirror symmetry

TL;DR

This paper explores mirror symmetry through quantum differential systems, focusing on the geometric case, and applies this framework to compute rational structures and identify mirror partners for specific orbifolds and surfaces.

Contribution

It introduces quantum differential systems as a framework for mirror symmetry, extending rational structure constructions to orbifolds and surfaces.

Findings

Computed the rational structure on orbifold cohomology of weighted projective spaces.

Identified a mirror partner for the Hirzebruch surface F2 using quantum differential systems.

Extended the rational structure framework to non-resonant, logarithmic cases.

Abstract

We study mirror symmetry (A-side vs B-side) in the framework of quantum differential systems. We focuse on the logarithmic and non-resonant case, which describes the geometric situation. We show that quantum differential systems provide a good framework in order to generalize the construction of the rational structure on the A-side given by Katzarkov, Kontsevitch and Pantev for the projective spaces. As an application, we compute the rational structure obtained in this way on the orbifold cohomology of weighted projective spaces. As an example we also calculate, using quantum differential systems, a mirror partner of the Hirzebruch surface F2.