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

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