# The Tropical Superpotential For $\mathbb{P}^2$

## TL;DR

This paper computes the tropical superpotential for the complement of a genus one plane curve, illustrating the wall and chamber structure and confirming the mirror symmetry predictions for .

## Contribution

It provides a detailed example of computing the tropical superpotential using the Gross--Siebert algorithm for a specific affine manifold.

## Key findings

- The superpotential matches the predicted Laurent polynomials for .
- The wall and chamber decomposition is explicitly determined.
- The superpotential is shown to be identical across chambers.

## Abstract

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, which we demonstrate to be identical to the Laurent polynomials predicted by Coates--Corti--Galkin--Golyshev--Kaspzryk to be mirror to $\mathbb{P}^2$.