On G-function of Frobenius manifolds related to Hurwitz spaces

Abstract

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.