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 $g\ge 2$ we obtain formulas for $|{\tau }_I{|}^2$ which allows to compute the real part of the $G$-function.