Riemann surfaces identifying between two tuples of data: complex structures, nodes, marked points and maps). Each of its component is compact and can be enumerated as $\overline{\mathcal{M}}_{g,m,\beta}(Q,\omega,J)$ for $g\geq 0, 3-2g-m\leq 0, \beta\in H_2(Q,\mathbb{Z})$. One needs to make $\overline{\mathcal{M}}(Q,\omega, J)$ transverse before one can define Gromov-Witten invariants by integration to satisfy Kontsevich-Manin axioms. The polyfold approach is to view $\overline{\mathcal{M}}(Q,\omega, J)$ inside the stable map space $B(Q,\omega)$ (see [@PolyfoldGW] 1.4), and $\overline{\mathcal{M}}(Q,\omega, J)$ is the solution space (in the orbit space) of a section $\bar\partial_J: B(Q,\omega)\to E(Q,\omega, J)$.

To describe a chart around $\alpha\in [\alpha]$ in $B(Q,\omega)$, one stabilizes the domain Riemann surface of $\alpha$ by adding extra marked points that are chosen with some rules to satisfy ([@PolyfoldGW] 3.1), describes the variation of newly marked and now stable domain Riemann surface using (reformulated) Deligne-Mumfold theory, then considers the variation of maps near the map part of $\alpha$ with codimension two transversal constraints at the images of those new marked points while allowing gluing of those maps at nodes. After specifying a tuple of data with properties, one can have a chart invariant under $G:=Aut(\alpha)$, called a **good uniformizer**, see [@PolyfoldGW] 3.9. Coordinate changes are included as special cases of 