TL;DR
This paper introduces a unified framework using dual fans to describe various mirror symmetry constructions, showing they are equivalent to pairs of toric Landau-Ginzburg models, thus unifying different approaches.
Contribution
It formalizes mirror constructions via dual fans and demonstrates their equivalence to toric Landau-Ginzburg models, unifying multiple mirror symmetry methods.
Findings
Dual fan models coincide with mirror pairs models.
Mirror constructions can be expressed as pairs of toric Landau-Ginzburg models.
The process is reversible, establishing an equivalence between models.
Abstract
We show that the mirror constructions of Greene-Plesser, Berglund-Hubsch, Batryev-Borsov, Givental and Hori-Vafa can be expressed in terms of what we call dual fans. To do this, we associate to a pair of dual fans a pair of toric Landau-Ginzburg models, and we describe a process by which each of the mirror constructions listed also produces a pair of toric Landau-Ginzburg models. Replacing mirror pairs by toric Landau-Ginzburg models is reversible, and our main result is the dual fan models and the mirror pairs models coincide.
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