  We construct a flat holomorphic line bundle over a connected component of the
Hurwitz space of branched coverings of the Riemann sphere. A flat holomorphic
connection defining the bundle is described in terms of the invariant Wirtinger
projective connection on the branched covering corresponding to a given
meromorphic function on a Riemann surface. In genera 0 and 1 we construct a
nowhere vanishing holomorphic horizontal section of this bundle (the
``Wirtinger tau-function''). In higher genus we compute the modulus square of
the Wirtinger tau-function.
