  We consider the factorization problem of matrix symbols relative to a closed
contour, i.e., a Riemann-Hilbert problem, where the symbol depends analytically
on parameters. We show how to define a function $\tau$ which is locally
analytic on the space of deformations and that is expressed as a Fredholm
determinant of an operator of "integrable" type in the sense of
Its-Izergin-Korepin-Slavnov. The construction is not unique and the
non-uniqueness highlights the fact that the tau function is really the section
of a line bundle.
