Intrinsic mirrors for minimal adjoint orbits and categories of singularities

TL;DR

This paper constructs and verifies mirror symmetry for Landau-Ginzburg models associated with minimal semisimple adjoint orbits of sl(n), demonstrating the Gross-Siebert recipe's effectiveness and proving Homological Mirror Symmetry for LG(2).

Contribution

It explicitly constructs mirror models for minimal adjoint orbits and proves Homological Mirror Symmetry for LG(2), advancing understanding of mirror symmetry in singularity categories.

Findings

Gross-Siebert recipe yields correct mirror objects.

Homological Mirror Symmetry proven for LG(2).

Landau-Ginzburg models have well-understood singularity categories.

Abstract

I discuss mirrors of Landau-Ginzburg models formed by a minimal semisimple adjoint orbit of $\mathfrak{sl}(n)$ together with a potential obtained via the Cartan-Killing form. I show that the Landau-Ginzburg models produced by the Gross-Siebert recipe give precisely the objects of the desired mirrors. It is known that Landau-Ginzburg model $LG(2)$ over the semisimple adjoint orbit of $\mathfrak{sl}(2)$ does not have projective mirrors. I prove Homological Mirror Symmetry for $LG(2)$ by constructing a Landau-Ginzburg mirror and showing that its Orlov category of singularities is equivalent to $Fuk(LG(2))$.