On G-function of Frobenius manifolds related to Hurwitz spaces

Abstract. The semisimple Frobenius manifolds related to the Hurwitz spaces Hg,N(k1, . . . , kl) are considered. We show that the corresponding isomonodromic tau-function τI coincides with (−1/2)power of the Bergmann tau-function which was introduced in a recent work by the authors [8]. This enables us to calculate explicitly the G-function of Frobenius manifolds related to the Hurwitz spaces H0,N (k1, . . . , kl) and H1,N(k1, . . . , kl). As simple consequences we get formulas for the G-functions of the Frobenius manifolds CN/W̃ (AN−1) and C×CN−1×{Iz > 0}/J(AN−1), where W̃ (AN−1) is an extended affine Weyl group and J(AN−1) is a Jacobi group, in particular, proving the conjecture of [13]. In case of Frobenius manifolds related to Hurwitz spaces Hg,N(k1, . . . , kl) with g ≥ 2 we obtain formulas for |τI |2 which allows to compute the real part of the G-function.