Given a log Calabi-Yau surface Y with maximal boundary D and distinguished complex structure, we explain how to construct a mirror Lefschetz fibration w:M→C, where M is a Weinstein four-manifold, such that the directed Fukaya category of w is isomorphic to DbCoh(Y), and the wrapped Fukaya category W(M) is isomorphic to DbCoh(Y∖D). We construct an explicit isomorphism between M and the total space of the almost-toric fibration arising in the work of Gross-Hacking-Keel; when D is negative definite this is expected to be the Milnor fibre of a smoothing of the dual cusp of D. We also match our mirror potential w with existing constructions for a range of special cases of (Y,D), notably in work of Auroux-Katzarkov-Orlov and Abouzaid.
Hacking et al. (Thu,) studied this question.