The nearby cycles of GKS kernels quantizing smoothed support-function Hamiltonians act on constructible sheaves by convolution with twisted polytope sheaves, matching the mirror Picard group action.
Abouzaid, Homogeneous coordinate rings and mirror symmetry for toric varieties, Geom
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Contact isotopies in the coherent-constructible correspondence
The nearby cycles of GKS kernels quantizing smoothed support-function Hamiltonians act on constructible sheaves by convolution with twisted polytope sheaves, matching the mirror Picard group action.