Consider $(\mathbb{ C}^*)^k$ acting on $\mathbb{ C}^N$ satisfying certain 'quasi-symmetric' condition which produces a class of toric Calabi-Yau GIT quotient stacks. Using subcategories of $Coh([\mathbb{ C}^N / (\mathbb{ C}^*)^k])$ generated by line bundles whose weights are inside certain zonotope called the 'magic window', Halpern-Leistner and Sam give a combinatorial construction of equivalences between derived categories of coherent sheaves for various GIT quotients. We apply the coherent-constructible correspondence for toric varieties to the magic windows and obtain a non-characteristic deformation of Lagrangian skeletons in $\mathbb{ R}^{ N-k}$ parameterized by $\mathbb{ R}^k$, exhibiting derived equivalences between A-models of the various phases. Moreover, by translating the magic window zonotope in $\mathbb{ R}^k$, we obtain a universal skeleton over $\mathbb{ R}^k \times \mathbb{ R}^k \setminus \mathcal{ D}$ for some fattening of hyperplane arrangements $\mathcal{ D}$, and we show that the the universal skeleton induces a local system of categories over $\mathbb{ R}^k \times \mathbb{ R}^k \setminus \mathcal{ D}$. We also connect our results to the perverse schober structure identified by \v{ S}penko and Van den Bergh.
Categories: math.SG