HMS symmetries of toric boundary divisors

Let X be a projective crepant resolution of a Gorenstein affine toric variety and let ( (C^*) 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 (C^*) ᵏ. Let KA be the so-called "stringy K\"ahler moduli space" of X. We show that ₁ (KA) acts on the wrapped Fukaya category of D. Using results by Gammage - Shende and Zhou, this result implies that ₁ (KA) acts on Dᵇ (coh (X) ) where X is the toric boundary divisor of X. We show that the induced action of ₁ (KA) on K₀ (coh (X) ) may be extended in a natural way to an action on K₀ (X) which corresponds to a GKZ system.