Donaldson-Thomas theory of $[{\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_{n+1}]\times {\mathbb{P}}^1$

We study the relative orbifold Donaldson-Thomas theory of $[{\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_{n+1}]\times {\mathbb{P}}^1$. We establish a correspondence between the DT theory relative to 3 fibers to quantum multiplication by divisors in the Hilbert scheme of points on $[{\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_{n+1}]$. This determines the whole theory if a further nondegeneracy condition is assumed. The result can also be viewed as a crepant resolution correspondence to the DT theory of $$\mathcal{A}$_n\times $\mathbb{P}$^1$.