Tropical Superpotential for $\mathbb{P}^2$

We present an extended worked example of the computation of the tropical superpotential considered by Carl–Pumperla–Siebert. In particular we consider an affine manifold associated to the complement of a non-singular genus one plane curve and calculate the wall and chamber decomposition determined by the Gross–Siebert algorithm. Using the results of Carl–Pumperla–Siebert we determine the tropical superpotential, via broken line counts, in every chamber of this decomposition. The superpotential defines a Laurent polynomial in every chamber, and we show that these are precisely the Laurent polynomials predicted by Coates–Corti–Galkin–Golyshev–Kaspzryk to be mirror to $\mathbb{P}^2$