STABLE PAIR THEORY ON TORIC ORBIFOLDS AND COLORED REVERSE PLANE PARTITIONS

We give a GIT construction for the moduli space of stable pairs on projective stacks, and study PT invariants on orbiflod toric Calabi-Yau threefolds with transverse $A_{n-1}$ singularities. The basic combinatorial object is the orbifold PT vertex $W^n_{\lambda\mu\nu}$. In the 1-leg case, $W^n_{\lambda\mu\nu}$ is the generating function for the number of $\mathbb{Z}_n$-colored reverse plane partitions, and we derive an explicit formula for $W^n_{\lambda\mu\nu}$ in terms of Schur functions. We also explicitly compute the PT partition function and verify the orbifold DT/PT correspondence for the local football $\operatorname{Tot}\left(\mathcal{O}(-p_0)\oplus\mathcal{O}(-p_\infty)\rightarrow\mathbb{P}^1_{a,b}\right)$