The Klemm–Mariño formula for Gromov–Witten invariants of Enriques surfaces

Let $Y$ be an Enriques surface. For genus $g$ and curve class $\beta$, let $N_{g,\beta}$ be the Hodge integral of the virtual class of the moduli space of degree-$\beta$ genus-$g$ stable maps to $Y$, and let $\omega_g(n)$ be defined by

$$\sum_{g \geq 0} \sum_{n \geq 0} \omega_g(n) (-1)^{g-1} z^{2g-2} q^n = \prod_{m \geq 1} \frac{ (1- e^z q^{2m})^2 (1- e^{-z} q^{2m})^2 (1-q^{2m})^4 }{ (1-e^z q^m)^2 (1- e^{-z} q^m)^2 (1-q^m)^{12} }.$$

Klemm–Mariño's formula. For all $g$ and $\beta$,

$$N_{g,\beta} = 2 \sum_{\substack{\operatorname{odd} k \mid \beta}} k^{2g-3} \omega_g\left( \frac{\beta^2}{2k^2} \right).$$

This is the conjectural explicit formula for the fundamental Gromov–Witten invariants of an Enriques surface. It was proved in genus $1$ only conditionally on conjectural Virasoro constraints, while the general formula remains open.