A Lie theoretical construction of a Landau-Ginzburg model without projective mirrors

TL;DR

This paper constructs a Landau-Ginzburg model using Lie theory and demonstrates that its Fukaya-Seidel category is equivalent to a subcategory of coherent sheaves on a Hirzebruch surface, showing no projective mirror exists.

Contribution

It provides a Lie theoretical construction of a Landau-Ginzburg model with a detailed categorical equivalence, highlighting the absence of projective mirrors.

Findings

Fukaya-Seidel category of LG(2) is equivalent to a subcategory of coherent sheaves on the Hirzebruch surface

No projective variety can serve as a mirror to LG(2)

This non-mirror property persists after compactification

Abstract

We describe the Fukaya-Seidel category of a Landau-Ginzburg model LG(2) for the semisimple adjoint orbit of sl(2,C). We prove that this category is equivalent to a full triangulated subcategory of the category of coherent sheaves on the second Hirzebruch surface. We show that no projective variety can be mirror to LG(2), and that this remains so after compactification.