Equivariant Gopakumar–Vafa correspondence for local $\mathbb{P}^1$

Let $X$ be the total space of the rank-two vector bundle

$$E\simeq\mathcal{O}_{\mathbb{P}^1}(k)\oplus\mathcal{O}_{\mathbb{P}^1}(-2-k)$$

on $\mathbb{P}^1$, and let $\beta=d[\mathbb{P}^1]\in H_2(X,\mathbb{Z})$. Write $N_d^{\rm GW}(k)$ for the genus-zero local Gromov–Witten invariant and $n_d(k)$ for the equivariant local BPS invariant defined by the residue integral.

Equivariant Gopakumar–Vafa correspondence. The Gopakumar–Vafa formula holds:

$$N_d^{\rm GW}(k)=\sum_{m\mid d}\frac{n_{d/m}(k)}{m^3}.$$

This predicts that the genus-zero Gromov–Witten invariants are related to the equivariant BPS invariants by the usual multiple-cover formula. The correspondence is motivated by earlier results for embedded contractible rational curves and ADE resolutions, while the local $\mathbb{P}^1$ case is the subject of this work.