The contraction-algebra BPS–Gopakumar–Vafa conjecture

Let $A$ be the contraction algebra of a rational flopping curve $C$, namely a Jacobi algebra of the type defined in the cited work. For every $d\in\mathbb{N}$, let $\Omega_{A,d}^{\operatorname{coh}}(q^{1/2})$ and $\Omega_{A,d}^{\operatorname{wt}}(q^{1/2})$ denote the refined BPS invariants defined using cohomological and weight refinements. Contraction-algebra BPS conjecture. The refined BPS invariants are constant and satisfy

$$\Omega_{A,d}^{\operatorname{coh}}(q^{1/2})=\Omega_{A,d}^{\operatorname{wt}}(q^{1/2}),$$

with this common value equal to the $d$th Gopakumar–Vafa invariant of $C$. This conjecture relates refined BPS invariants of contraction algebras to curve-counting invariants arising from rational flops.