Stable-pairs rationality conjecture for descendent partition functions

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Stable pairs rationality conjecture for descendent partition functions

   Let $X$) be a nonsingular projective $3$-fold, let $0\ne\beta\in H_2(X,\mathbb{Z})$, and let $\gamma_i\in H^*(X,\mathbb{Q})$. Write $\mathsf Z_{\mathsf P}\big(X;q\ \big|\prod_{i=1}^r\tau_{k_i}(\gamma_i)\big)_\beta$ for the stable pairs descendent partition function.

   Stable pairs rationality conjecture. The partition function $\mathsf Z_{\mathsf P}\big(X;q\ \big|\prod_{i=1}^r\tau_{k_i}(\gamma_i)\big)_\beta$ is the Laurent expansion of a rational function in $q$.

   This rationality statement is the foundational generating-series assertion underlying the stable-pairs side of the GW/Pairs correspondence. The source attributes it to earlier work, but no resolution status is supplied here.

   source: R. Pandharipande and A. Pixton, “Gromov-Witten/Pairs correspondence for the quintic 3-fold”, arXiv:1206.5490 (2016).