Monodromy action of mirror stops for toric Calabi-Yau surfaces

Mirror symmetry predicts an action by the fundamental group of a conjectural stringy Kähler moduli space on the derived category of an algebraic variety. For a toric variety, a model for this space is understood, but constructing the action is still an open problem in general. We propose that this action can be studied on the $A$-side via a moduli space of Legendrians isotopic to the FLTZ Legendrian. For the $A_{n-1}$ singularity, we construct an annular braid-group action on the corresponding partially wrapped Fukaya category by exact autoequivalences. The standard braid subgroup recovers the Seidel–Thomas action on the derived category, while the additional annular generator corresponds to tensor product with $\mathcal{O}(-1)$. We additionally extend the Floer-theoretic approach to homological mirror symmetry for toric varieties to the setting of semiprojective toric Deligne–Mumford stacks over an arbitrary field.