The relative GW/PT correspondence

Let $(X,D)$ be the relative geometry, let $\beta$ be a curve class, let $\lambda$ be a relative boundary condition, and let $\gamma_1,\ldots,\gamma_r\in H^{\ast}(X)$. Write $\overline{\tau_{\alpha_1-1}(\gamma_1)\cdots\tau_{\alpha_r-1}(\gamma_r)}$ for the relative descendent transformation defined using the universal correspondence matrix.

The relative GW/PT correspondence. $Z^{(X,D)}_{\mathsf{PT},\beta}(\lambda\mid\tau_{\alpha_1-1}(\gamma_1)\cdots\tau_{\alpha_r-1}(\gamma_r))$ is the Fourier expansion of a rational function in $p$, and under $p=e^z$,

$$Z^{(X,D)}_{\mathsf{PT},\beta}(\lambda\mid\tau_{\alpha_1-1}(\gamma_1)\cdots\tau_{\alpha_r-1}(\gamma_r))=Z^{(X,D)}_{\mathsf{GW},\beta}(\lambda\mid\overline{\tau_{\alpha_1-1}(\gamma_1)\cdots\tau_{\alpha_r-1}(\gamma_r)}).$$

This is the relative version of the GW/PT correspondence, extending rationality and the GW/PT identification to marked relative insertions.