Crepant resolution conjecture for local projective four-space and $[\mathbb{C}^5/\mathbb{Z}_5]$

Let $K\mathbb{P}^4$ be the total space of the canonical bundle over $\mathbb{P}^4$, and let $[\mathbb{C}^5/\mathbb{Z}_5]$ be the corresponding orbifold. Write $\mathsf{R}^{K\mathbb{P}^4}$ and $\mathsf{R}^{[\mathbb{C}^5/\mathbb{Z}_5]}$ for their respective $\mathsf{R}$-matrices. Let $\mathds{T}$ be the transformation from the coefficient ring of $K\mathbb{P}^4$ to that of $[\mathbb{C}^5/\mathbb{Z}_5]$ defined by

$$\mathds{T}(L)=-\frac{L}{5},\qquad \mathds{T}(X)=-\frac{X}{5},\qquad \mathds{T}(DX)=\frac{DX}{5^2},\qquad \mathds{T}(D^2X)=-\frac{D^2X}{5^3},\qquad \mathds{T}(Y)=-\frac{Y}{5}.$$

Crepant resolution conjecture. The transformation $\mathds{T}$ satisfies

$$\mathds{T}\left(\mathsf{R}^{K\mathbb{P}^4}\right)=\mathsf{R}^{[\mathbb{C}^5/\mathbb{Z}_5]}.$$

This is the explicit all-genus crepant resolution conjecture relating the Gromov–Witten theories of local $\mathbb{P}^4$ and $[\mathbb{C}^5/\mathbb{Z}_5]$. The paper proves the conjecture for genera $2$ and $3$, while the all-genus statement remains open in the supplied text.