The semisimple Frobenius manifolds related to the Hurwitz spaces $H_{ g,N}(k_1, ..., k_l)$ are considered. We show that the corresponding isomonodromic tau-function $\tau_I$ coincides with $(-1/2)$-power of the Bergmann tau-function which was introduced in a recent work by the authors \cite{ KokKor}. This enables us to calculate explicitly the $G$-function of Frobenius manifolds related to the Hurwitz spaces $H_{ 0, N}(k_1, ..., k_l)$ and $H_{ 1, N}(k_1, ..., k_l)$. As simple consequences we get formulas for the $G$-functions of the Frobenius manifolds ${ \mathbb C}^N/\tilde{ W}^k(A_{ N-1})$ and ${ \mathbb C}\times{ \mathbb C}^{ N-1}\times\{ \Im z >0\}/J(A_{ N-1})$, where $\tilde{ W}^k(A_{ N-1})$ is an extended affine Weyl group and $J(A_{ N-1})$ is a Jacobi group, in particular, proving the conjecture of \cite{ Strachan}. In case of Frobenius manifolds related to Hurwitz spaces $H_{ g, N}(k_1, ..., k_l)$ with $g\geq2$ we obtain formulas for $|\tau_I|^2$ which allows to compute the real part of the $G$-function.
Categories: math-ph math.MP nlin.SI