arXiv:math-ph/0310008v1 7 Oct 2003 Bergmann tau-function on Hurwitz spaces and its applications A. Kokotov ∗ and D. Korotkin † October 28, 2018 Max-Planck Institute of Mathematics Vivatsgasse 7, D-53111 Bonn, Germany Abstract. The main result of this work is a computation of the Bergmann tau-function on Hurwitz spaces in any genus. This allows to get an explicit formula for the G-function of Frobenius manifolds associated to arbitrary Hurwitz spaces, get a new expression for determinant of Laplace operator in Poincar´e metric on Riemann surfaces of arbitrary genus, and compute Jimbo-Miwa tau-function of an arbitrary Riemann-Hilbert problem with quasi-permutation monodromies. 1 Introduction The Hurwitz space H g,N is the space of equivalence classes of pairs (L, π), where L is a compact Riemann surface of genus g and π is a meromorphic function of degree N . The Hurwitz space is stratified according to multiplicities of poles and critical points of function π (see [19, 9]); in this paper we shall mainly work within the generic stratum H g,N (1,..., 1), for which all critical points and poles of function π are simple. Denote the critical points of function π by P 1 ,...P M (M =2N +2g −2 according to the Riemann-Hurwitz formula); the critical values λ m = π(P m ) can be used as (local) coordinates on H g,N (1,..., 1). The function π defines a realization of the Riemann surface L as an N -sheeted branched covering of CP 1 with ramification points P 1 ,...,P M and branch points λ m = π(P m ); enumerate the points at infinity of the branched covering in some order and denote them by ∞ 1 ,..., ∞ N . In a neighbourhood of the ramification point P m the local coordinate is chosen to be x m (P )=  π(P ) − λ m , m =1,...,M ; in a neighbourhood of any point ∞ n the local parameter is x M+n (P )=1/π(P ), n =1,...,N . Fix a canonical basis of cycles (a α ,b α ) on L and introduce the prime-form E(P,Q) on L and Bergmann bidifferential B(P,Q)= d P d Q ln E(P,Q) (a symmetric bimeromorphic differential with zero a-periods). The Bergmann bidifferential has the second order pole as Q → P with the following local behaviour: B(P,Q)/{dx(P )dx(Q)} =(x(P ) − x(Q)) −2 + 1 6 S B (x(P ))+ o(1), where x(P ) is a local coordinate; S B (x(P )) is the Bergmann projective connection. We define the Bergmann τ -function τ (λ 1 ,...,λ M ) locally by the system of equations ∂ ∂λ m ln τ = − 1 12 S B (x m )| xm=0 , m =1,...,M ; (1.1) * e-mail: alexey@mathstat.concordia.ca † e-mail: korotkin@mathstat.concordia.ca; on leave of absence from Department of Mathematics and Statistics, Con- cordia University 7141 Sherbrooke West, Montreal H4B 1R6, Quebec, Canada 1