Lee's spin Gromov–Witten/Hurwitz correspondence

Let $(B,\vartheta)$ be a spin curve, let $g,d$ be integers, and let $k_1,\ldots,k_n$ satisfy

$$\sum_i k_i=g-1+d(1-g(B))+n.$$

Write $\overline{c}_k=\sum_{\mu\ \textup{odd}}\kappa_{k,\mu}\cdot\mu$ for the spin completed cycle, let $H_d(B,\vartheta;\mu^1,\ldots,\mu^n)$ denote the spin Hurwitz number, and define the stationary spin Gromov–Witten invariants by

$$\left\langle \tau_{k_1}\cdots\tau_{k_n}\right\rangle_{g,d}^{B,\vartheta}=\int_{[\overline{\mathcal{M}}_{g,n}(B,d)]^{{\rm loc},\vartheta}}\prod_{i=1}^n\psi_i^{k_i}\operatorname{ev}_i^*(\omega),$$

where $\omega\in H^2(\mathbb{P}^1,\mathbb{Z})$ is the class of a point. Spin GW/H correspondence. For all such data,

$$\left\langle \tau_{k_1}\cdots\tau_{k_n}\right\rangle_{g,d}^{B,\vartheta}=H_d\left(B,\vartheta;\frac{(-1)^{k_1}k_1!}{(2k_1)!}\overline{c}_{2k_1+1},\ldots,\frac{(-1)^{k_n}k_n!}{(2k_n)!}\overline{c}_{2k_n+1}\right),$$

with $H_d$ extended by linearity. The conjecture generalises earlier conjectures of Maulik–Pandharipande; it is known for $(B,\vartheta)=(\mathbb{P}^1,\mathcal{O}(-1))$, while the general correspondence remains open.