The half Seiberg–Witten curve topological free-energy conjecture

Let $r>2$ and let $Q_1,\ldots,Q_r$ and $\Lambda$ be the parameters of the half Seiberg–Witten curve, with $B_{2g}$ denoting the Bernoulli number. Topological free-energy conjecture. For the half Seiberg–Witten curve, the topological free energies are

$$F_0=\sum_{1\leq a<b\leq r}\left[\frac{3}{4}(Q_a-Q_b)^2+\frac{1}{4}(Q_a-Q_b)^2\ln\left(\frac{\Lambda^{r}}{(Q_a-Q_b)^2}\right)\right],$$ $$F_1=-\sum_{1\leq a<b\leq r}\frac{1}{24}\ln\left(\frac{\Lambda^{r}}{(Q_a-Q_b)^2}\right),$$

and, for $g\geq2$,

$$F_g=-\frac{B_{2g}}{2g(2g-2)}\sum_{1\leq a<b\leq r-1}(Q_a-Q_b)^{2-2g}.$$

These formulae provide the proposed closed expressions for the topological free energies in the half Seiberg–Witten-curve case, where analogous formulae for the Gaiotto and CDO curves are not available for $r>2$.