GKZ discriminant and Multiplicities

TL;DR

The paper investigates a numerical facet of homological mirror symmetry for toric Calabi–Yau GIT problems, showing that A-side tropical discriminant multiplicities along GKZ walls match B-side semi-orthogonal decomposition multiplicities of toric wall-crossing. The approach hinges on GKZ tropicalization, minimal faces, and a Coulomb–Higgs framework that yields a key lemma: SOD multiplicities are invariant under passing to Coulomb problems. Leveraging Horja–Katzarkov’s discriminant results and a recursive rank formula for K_0, the authors prove n^A_{W,F} = n^B_{W,F} for all codimension-1 walls W and minimal faces F, and express multiplicities through stacky volumes. This bridges A- and B-model data, provides a practical recursion to compute multiplicities via K-theory ranks, and strengthens the conceptual link between tropical and categorical invariants in toric CY settings.

Abstract

Let $T=(\C^*)^k$ act on $V=\C^N$ faithfully and preserving the volume form, i.e. $(\C^*)^k \into \text{SL}(V)$. On the B-side, we have toric stacks $Z_W$ (see Eq. \ref{eq:ZW})labelled by walls $W$ in the GKZ fan, and $Z_{/F}$ labelled by faces of a polytope corresponding to minimal semi-orthogonal decomposition (SOD) components. The B-side multiplicity $n^B_{W,F}$, well-defined by a result of Kite-Segal \cite{kite-segal}, is the number of times $\Coh(Z_{/F})$ appears in a complete SOD of $\Coh(Z_W)$. On the A-side, we have the GKZ discriminant loci components $\nabla_F \In (\C^*)^k$, and its tropicalization $\nabla^{trop}_{F} \In \R^k$. The A-side multiplicity $n^A_{W, F}$ is defined as the multiplicity of the tropical complex $\nabla^{trop}_{F}$ on wall $W$. We prove that $n^A_{W,F} = n^B_{W,F}$, confirming a conjecture in Kite-Segal \cite{kite-segal} inspired by \cite{aspinwall2017mirror}. Our proof is based on the result of Horja-Katzarkov \cite{horja2022discriminants} and a lemma about B-side SOD multiplicity, which allows us to reduce to lower dimension just as in A-side \cite{GKZ-book}[Ch 11].

Figures (3)

• Figure 1: Tropicalization gives balanced polyhehral complex. The integers labels multiplicities on each cell.
• Figure 2: Multiplicities from Volume allocation.
• Figure 3: Decomposition of $\nabla^B_{GKZ} = \sum_H \nabla^B_H$, where $H$ labels the relevant subspaces.

Theorems & Definitions (42)

• Example 1.1
• Theorem 1.2: Lemma \ref{['l:B-pullback']}
• Theorem 1.3: Theorem \ref{['t: main']}
• Theorem 1.4: Proposition \ref{['p:recursive']}
• Remark 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Remark 2.4
• Remark 2.5