Homological Mirror Symmetry for Conic Bundle

We study the homological mirror symmetry statement where A-side is the conic bundle Hori–Vafa mirror $\mathcal{Y}=\{uv=f(z)\}\subset {\mathbb{C}}^2\times ({\mathbb{C}}^{\ast }{)}^n$ for a Laurent polynomial $f$ in $({\mathbb{C}}^{\ast }{)}^n$, and B-side is some a toric Calabi–Yau $(n+2)$-fold with a smooth anti-canonical divisor removed ${\mathcal{X}}^{\circ }=\mathcal{X}\setminus w^{-1}(-1)$. We show that when $\mathcal{X}$ is the canonical bundle of a toric Fano $n$-orbifold $S$ and $f$ is its Givental superpotential, the strong deformation retraction skeleton $\mathsf{L}$ of $\mathcal{Y}$ in the sense of RSTZ (Ruddat–Sibilla–Treumann–Zaslow in Geom. Topol. 18(3):1343–1395, 2014) has a Weinstein neighborhood $U$, such that the wrapped microlocal sheaf category $\mu {\mathrm{S}\mathrm{h}}_{\mathsf{L}}^w(\mathsf{L})\cong \mathrm{C}\mathrm{o}\mathrm{h}({\mathcal{X}}^{\circ })$. This proves a microlocal categorical version of the SYZ mirror in (Abouzaid–Auroux–Katzarkov in Publ. math. IHÉS 123(1):199–282, 2016, Thm. 1.7). We also extend the definition of characteristic cycles for constructible sheaves in cotangent bundles from (Kashiwara–Schapira in Sheaves on Manifolds, Grundlehren math. Wiss. 292, Springer, 1990, Ch. IX) to finite-rank objects in $\mu {\mathrm{S}\mathrm{h}}_{\mathsf{L}}^w(\mathsf{L})$, and describe the characteristic cycles for objects mirror to a coherent sheaf supported on $S$.