HMS symmetries of toric boundary divisors

Abstract

Let $X$ be a projective crepant resolution of a Gorenstein affine toric variety and let $(({\mathbb{C}}^{\ast }{)}^k,f)$ be the LG-model which is the Hori-Vafa mirror dual of $X$. Let $D$ be a generic fiber of $f$ equipped with the restriction of the standard Liouville form on $({\mathbb{C}}^{\ast }{)}^k$. Let ${\mathcal{K}}_A$ be the so-called "stringy Kähler moduli space" of $X$. We show that ${\pi }_1({\mathcal{K}}_A)$ acts on the wrapped Fukaya category of $D$. Using results by Gammage - Shende and Zhou, this result implies that ${\pi }_1({\mathcal{K}}_A)$ acts on $D^b(\mathrm{coh}(\partial X))$ where $\partial X$ is the toric boundary divisor of $X$. We show that the induced action of ${\pi }_1({\mathcal{K}}_A)$ on $K_0(\mathrm{coh}(\partial X))$ may be extended in a natural way to an action on $K_0(X)$ which corresponds to a GKZ system.

75 pages