GKZ discriminant and Multiplicities

Let $T=({\text{C}}^{\ast }{)}^k$ act on $V={\text{C}}^N$ faithfully and preserving the volume form, i.e. $({\text{C}}^{\ast }{)}^k\text{into}\text{SL}(V)$. On the B-side, we have toric stacks $Z_W$ (see Eq. \ref{eq:ZW})labelled by walls $W$ in the GKZ fan, and $Z_{\mathrm{/}F}$ labelled by faces of a polytope corresponding to minimal semi-orthogonal decomposition (SOD) components. The B-side multiplicity $n_{W,F}^B$, well-defined by a result of Kite-Segal \cite{kite-segal}, is the number of times $\text{Coh}(Z_{\mathrm{/}F})$ appears in a complete SOD of $\text{Coh}(Z_W)$. On the A-side, we have the GKZ discriminant loci components ${\mathrm{\nabla }}_F\text{In}({\text{C}}^{\ast }{)}^k$, and its tropicalization ${\mathrm{\nabla }}_F^{trop}\text{In}{\mathbb{R}}^k$. The A-side multiplicity $n_{W,F}^A$ is defined as the multiplicity of the tropical complex ${\mathrm{\nabla }}_F^{trop}$ on wall $W$. We prove that $n_{W,F}^A=n_{W,F}^B$, 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].