The tropical superpotential for $\mathbb{P}^2$
Thomas Prince · Algebraic geometry · 2019
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 with the complement of a non-singular genus 1 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 P 2 .