Intrinsic mirrors for minimal adjoint orbits and categories of singularities

TL;DR

This work develops and applies the Gross–Siebert intrinsic mirror symmetry program to minimal semisimple adjoint orbits of $\mathfrak{sl}(n)$, producing Landau–Ginzburg mirror models $({\mathcal{Y}}_{n+1},g)$ whose Orlov categories of singularities $D_{sg}(\mathcal{Y}_{n+1})$ reproduce the Fukaya–Seidel categories of the corresponding A-model LGs. It provides explicit realizations for $n=2$ (LG$(2)$) and $n=3$ (LG$(3)$) and generalizes to LG$(n+1)$, detailing compactifications, critical fibers, vanishing cycles, and the associated derived-category structures. In particular, HMS is established in the LG$(2)$ case by identifying a two-object $D_{sg}$-category isomorphic to the Fukaya–Seidel category, while higher $n$ cases yield mirror candidates with fully described object-generation in $D_{sg}$. The results offer a coherent framework linking log-geometry, tropicalization data, and derived categories for mirrors of minimal adjoint-orbit LG models, with potential implications for broader classes of adjoint-orbit LGs and their categorical mirrors.

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 $Fuk(LG(2))$.

Figures (2)

• Figure 1: $\mathop{\mathrm{LG}}\nolimits_2$ and its mirror
• Figure 2: $\mathop{\mathrm{LG}}\nolimits(3)$ and its mirror

Theorems & Definitions (32)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem
• Remark 4
• Lemma 5
• proof
• Lemma 6
• proof
• Theorem : \ref{['mirror2']}