Contact isotopies in the coherent-constructible correspondence

TL;DR

This paper geometrically realizes the mirror Picard group action in the coherent-constructible correspondence using contact isotopies, linking Hamiltonian flows on the cotangent bundle to sheaf-theoretic operations that mirror line bundle actions.

Contribution

It provides a sheaf-theoretic interpretation of the mirror Picard group action via quantized contact isotopies, connecting Hamiltonian flows to sheaf convolution in toric mirror symmetry.

Findings

The kernel $K_0$ corresponds to the Hamiltonian flow of the support function of $D$.

The action of $K_0$ matches the convolution with the twisted polytope sheaf.

This mirrors the action of line bundles on coherent sheaves in the mirror symmetry context.

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 $H_\varepsilon$ on $T^* T^n$. Their flows act on sheaves via a family of kernels $K_\varepsilon$ on $T^n \times T^n$. The nearby cycles kernel $K_0$ corresponds heuristically to the Hamiltonian flow of the non-differentiable function $\lim_{\varepsilon \to 0} H_\varepsilon$, 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.