GKZ discriminant and Multiplicities

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].