Rationality and symmetry conjecture for stable-pair generating series

Let $X$ be a smooth projective $3$-fold, let $\beta$ be an effective curve class, and let

$$D=\operatorname{ch}_{k_1}(\gamma_1)\ldots\operatorname{ch}_{k_n}(\gamma_n)$$

be a formal product of tautological classes. Write

$$Z_\beta^\mathsf{PT}(q\mid D)=\sum_m q^m\int_{[P_{\beta,m}]^\mathrm{vir}}D.$$

Stable-pair rationality and symmetry conjecture. The series $Z_\beta^\mathsf{PT}(q\mid D)$ is the Laurent expansion of a rational function, more precisely of $q^{d_\beta/2}f(q)$ for a rational function $f(q)$, and satisfies

$$Z_\beta^\mathsf{PT}(q^{-1}\mid D)=(-1)^{k_1+\ldots+k_n}Z_\beta^\mathsf{PT}(q\mid D).$$

This conjecture extends the rationality and functional-equation phenomena for stable-pair generating series from Calabi--Yau threefolds to general smooth projective threefolds. The paper presents these properties as expected in the general case; their full validity remains open.