The GW/PT correspondence for absolute threefolds

Let $X$ be a smooth projective threefold, let $\beta$ be a curve class, let $\boldsymbol{\tau}_{\alpha-1}(\boldsymbol{\gamma})=\tau_{\alpha_1-1}(\gamma_1)\cdots\tau_{\alpha_r-1}(\gamma_r)$ for a partition $\alpha=(\alpha_1,\ldots,\alpha_r)$ and classes $\gamma_i\in H^{\ast}(X)$, and let $\overline{\boldsymbol{\tau}_{\alpha-1}(\boldsymbol{\gamma})}$ denote the descendent transformation defined using the universal correspondence matrix. The partition functions $Z^{X}_{\mathsf{PT},\beta}$ and $Z^{X}_{\mathsf{GW},\beta}$ are formed from the PT and GW invariants, respectively.

The GW/PT correspondence. $Z^{X}_{\mathsf{PT},\beta}(\boldsymbol{\tau}_{\alpha-1}(\boldsymbol{\gamma}))$ is the Fourier expansion of a rational function in $p$, and under $p=e^z$,

$$Z^{X}_{\mathsf{PT},\beta}(\boldsymbol{\tau}_{\alpha-1}(\boldsymbol{\gamma}))=Z^{X}_{\mathsf{GW},\beta}(\overline{\boldsymbol{\tau}_{\alpha-1}(\boldsymbol{\gamma})}).$$

This asserts rationality of the PT partition function and identifies it, after the descendent transformation and the change of variables $p=e^z$, with the GW partition function.