Stacky GKM Graphs and Orbifold Gromov-Witten Theory

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Following Zong (arXiv:1604.07270), we define an algebraic GKM orbifold $\mathcal{X}$ to be a smooth Deligne-Mumford stack equipped with an action of an algebraic torus $T$, with only finitely many zero-dimensional and one-dimensional orbits. The 1-skeleton of $\mathcal{X}$ is the union of its zero-dimensional and one-dimensional $T$-orbits; its formal neighborhood $\hat{\mathcal{X}}$ in $\mathcal{X}$ determines a decorated graph, called the stacky GKM graph of $\mathcal{X}$. The $T$-equivariant orbifold Gromov-Witten (GW) invariants of $\mathcal{X}$ can be computed by localization and depend only on the stacky GKM graph of $\mathcal{X}$ with the $T$-action. We also introduce abstract stacky GKM graphs and define their formal equivariant orbifold GW invariants. Formal equivariant orbifold GW invariants of the stacky GKM graph of an algebraic GKM orbifold $\mathcal{X}$ are refinements of $T$-equivariant orbifold GW invariants of $\mathcal{X}$.