Quiver formula for the extended Goldman Poisson bracket

Following Fock and Goncharov, consider a triangulation $\Sigma_0$ and its subtriangulation into sub-triangles, with the associated extended quiver $\mathscr Q$. The nodes of $\mathscr Q$ carry the Fock–Goncharov variables and the additional toric variables, and let $\sigma,\sigma'$ be the logarithms of any two variables associated with nodes $p,p'$ of this quiver. The symbol $\epsilon_{p,p'}$ is determined by the arrow between the nodes: it is $1$ when the arrow goes from $p$ to $p'$, $-1$ in the opposite direction, and $\pm \frac12$ for a dashed arrow. Quiver formula for the extended Goldman Poisson bracket. The Poisson bracket inverse to the extended Goldman symplectic form $\mathcal W$ satisfies

$$n^2\{\sigma,\sigma'\}=\epsilon_{p,p'}.$$

In particular, the brackets between Fock–Goncharov variables are given by the original Fock–Goncharov quiver. This extends the usual quiver description of the Goldman structure to the added toric variables associated with cherries; the parser supplies no evidence resolving the assertion, so its status remains open.