Abstract

We provide a proof of the form taken by the Seiberg-Witten tau-function on the Hurwitz space of N-fold ramified covers of the Riemann sphere by a compact Riemann surface of genus g, a result derived in [10] for a special class of monodromy data. To this end we examine the Riemann-Hilbert problem with N×N quasi-permutation monodromies, whose corresponding isomonodromic tau-function contains the Seiberg-Witten tau-function as one of three factors. We present the solution of the Riemann-Hilbert problem following [11]. Along the way we give elementary proofs of variational formulas on Hurwitz spaces, including the Rauch formulas.