The BPS polynomial and K-theoretic refined BPS correspondence

Let $S$ be a smooth projective surface, let $\beta\in H_2(S,\mathbb{Z})$ satisfy

$$m_\beta=-1+c_1(S)\cdot\beta\geq 0,$$

and let $K_S$ be the total space of the canonical line bundle over $S$, viewed as a non-compact Calabi–Yau $3$-fold. Denote by $BPS_\beta^S(q)$ the BPS polynomial of $S$, and by $\Omega_\beta^{K_S}(q)$ the refined genus-zero BPS invariant of $K_S$ with $m_\beta$ point insertions. BPS–refined-BPS conjecture.

$$BPS_\beta^S(q)=\Omega_\beta^{K_S}(q).$$

This proposes an interpretation of the surface BPS polynomials through K-theoretic refined BPS invariants of the local Calabi–Yau geometry $K_S$; the conjecture is presented without a resolution in the source.