Introduced by Maulik-Okounkov-Nekrasov-Pandharipande,the Gromov-Witten/Donaldson-Thomas correspondence states that the generating functions of the two curve counting theories can be related to each other by a change of variables.This talk is aimed at a generalization of this correspondence to the case of local gerby curves with transversal A_n-singularity.The degeneration formulas for GW and DT theories enable us to organize those invariants into 2d topological quantum field theories,and reduce the correspondence to certain base cases.In the base case of [C^2/Z_{n+1}] \times P^1,our results actually establish a triangle of equivalent theories: GW theory,DT theory and the quantum cohomology of Hilbert scheme of points,which can be viewed as a crepant resolution counterpart to the triangle for A_n \times P^1 in the work of Maulik-Oblomkov.This is joint work with Zhengyu Zong.