Cobordism rationality and functional equation conjecture for stable-pair partition functions

Let $X$ be a nonsingular projective 3-fold, let $\beta eq 0$ be a curve class, and let $d=\int_\beta c_1(X)$ be the virtual dimension. Define the stable-pair cobordism partition function by

$$Z_{\Omega_\ast,\beta}^{X}(q)=\sum_n [P_{n,\beta}]q^n\in \Omega_\ast(\mathrm{pt})\otimes\mathbb{Q}((q)).$$

After choosing a basis $\{v_I\}$ of $\Omega_d(\mathrm{pt})_\mathbb{Q}$, write

$$Z_{\Omega_\ast,\beta}^{X}(q)=\sum_I f_I(q)\cdot v_I.$$

Cobordism rationality and functional equation conjecture. The function $Z_{\Omega_\ast,\beta}^{X}(q)$ is a rational function in $q$; more precisely, every $f_I(q)$ is the Laurent expansion of a rational function in $q$. Moreover,

$$Z_{\Omega_\ast,\beta}^{X}(q^{-1})=q^{-d}Z_{\Omega_\ast,\beta}^{X}(q),$$

so each $f_I(q)$ satisfies the same functional equation.

These properties extend the rationality and functional equation expected for stable-pair and Donaldson–Thomas partition functions to cobordism-valued invariants. The parser supplies no evidence that this conjecture has been resolved.