The Moduli Space of Harnack Curves in Toric Surfaces

In 2006, Kenyon and Okounkov computed the moduli space of Harnack curves of degree $d$ in $\mathbb{C}\mathbb{P}^2$. We generalize to any projective toric surface some of the techniques used there. More precisely, we show that the moduli space $\mathcal{H}_\Delta$ of Harnack curves with Newton polygon $\Delta$ is diffeomorphic to $\mathbb{R}^{m-3}\times\mathbb{R}_{\geq0}^{n+g-m}$ where $\Delta$ has $m$ edges, $g$ interior lattice points and $n$ boundary lattice points, solving a conjecture of Cr\'etois and Lang. Additionally, we use abstract tropical curves to construct a compactification of this moduli space by adding points that correspond to collections of curves that can be patchworked together to produce a curve in $\mathcal{H}_\Delta$. This compactification comes with a natural stratification with the same poset as the secondary polytope of $\Delta$.