element $\lambda$ of the torus ${\mathbb{C}}^*$ acts by $$\lambda \cdot (x_0, \ldots, x_n) = (\lambda^{q_0} x_0, \ldots, \lambda^{q_n} x_n).$$ The quotient stack $W{\mathbb{P}}^n(q_0, \ldots, q_n)/ \tilde G$ is equivalent to the stack $$\left[{\mathbb{C}}^{n+1}\setminus\{0\} / G{\mathbb{C}}^*\right]$$ so we can view the Calabi-Yau orbifold $Z_{A, G}$ as the (smooth) DM stack $$[Z_{A,G}] : = \left[ \left\{ x \in {\mathbb{C}}^{n+1} \setminus \{0\} | F_A(x) = 0\right\} /G{\mathbb{C}}^*\right] \subseteq \left[{\mathbb{C}}^{n+1}\setminus\{0\} / G{\mathbb{C}}^*\right].$$ We now review the Chen-Ruan orbifold cohomology for such a stack. Intuitively speaking, it consists of a direct sum over all elements of $G{\mathbb{C}}^*$ of $G$-invariant cohomology of the fixed loci of each element.

If $\gamma$ is an element of $G{\mathbb{C}}^*$, take the fixed loci $$\begin{aligned}
 {\mathbb{C}}_\gamma^{n+1}:&= \{ \mathbf{x} \in {\mathbb{C}}^{n+1}\setminus \{0\} | \gamma \cdot\mathbf{x} = \mathbf{x}\};\text{ and }  \\ X_A^\gamma :&= \{ F_A|_{{\mathbb{C}}_\gamma^{n+1}} = 0\} \subset {\mathbb{C}}_\gamma^{n+1}.
 \end{aligned}$$

Fix a point $\mathbf{x} \in X_A^\gamma$. The action of $\gamma$ on the tangent space $T_{\mathbf{x}}(\{F_A = 0\})$ can be written as a diagonal matrix (when written with respect to a certain basis), $\Lambda_\gamma = {\text{diag}}(e^{2\pi i a_1^\gamma}, \ldots, e^{2\pi i a_{n}^\gamma})$, for some real numbers $a_i^\gamma \in [0,1)$. We then define the *age shift* of $\gamma$, $$a(\gamma) := \frac{1}{2\pi i} \log(\det \Lambda_\gamma) = 