The proper Landau-Ginzburg potential is the open mirror map

The mirror dual of a smooth toric Fano surface X equipped with an anticanonical divisor E is a Landau–Ginzburg model with superpotential, W. Carl–Pumperla–Siebert give a definition of the superpotential in terms of tropical disks [13] using a toric degeneration of the pair (X,E). When E is smooth, the superpotential is proper. We show that this proper superpotential equals the open mirror map for outer Aganagic–Vafa branes in the canonical bundle K_X, in framing zero. As a consequence, the proper Landau–Ginzburg potential is a solution to the Lerche–Mayr Picard–Fuchs equation. Along the way, we prove a generalization of a result about relative Gromov–Witten invariants by Cadman–Chen to arbitrary genus using the multiplication rule of quantum theta functions. In addition, we generalize a theorem of Hu that relates Gromov–Witten invariants of a surface under a blow-up from the absolute to the relative case. One of the two proofs that we give introduces birational modifications of a scattering diagram. We also demonstrate how the Hori–Vafa superpotential is related to the proper superpotential by mutations from a toric chamber to the unbounded chamber of the scattering diagram.