Contact isotopies in the coherent-constructible correspondence

Abstract

The coherent-constructible correspondence is a realization of toric mirror symmetry in which the A-side is modeled by constructible sheaves on $T^n$. This paper provides a geometric realization of the mirror Picard group action in this correspondence, characterizing it in terms of quantized contact isotopies and providing a sheaf-theoretic counterpart to work of Hanlon in the Fukaya-Seidel setting. Given a toric Cartier divisor $D$, we consider a family of homogeneous Hamiltonians $H_{\epsilon }$ on $T^{\ast }{T}^n$. Their flows act on sheaves via a family of kernels $K_{\epsilon }$ on $T^n\times T^n$. The nearby cycles kernel $K_0$ corresponds heuristically to the Hamiltonian flow of the non-differentiable function ${\lim }_{\epsilon \to 0}{H}_{\epsilon }$, which is the pullback of the support function of $D$ along the cofiber projection. We show that the action of $K_0$ coincides with the convolution action of the associated twisted polytope sheaf, hence mirrors the action of $\mathcal{O}(D)$ on coherent sheaves.

arXiv comment26 pages