  We study families of quantum field theories of free bosons on a compact
Riemann surface of genus g. For the case g > 0 these theories are parameterized
by holomorphic line bundles of degree g-1, and for the case g=0 - by smooth
closed Jordan curves on the complex plane. In both cases we define a notion of
tau-function as a partition function of the theory and evaluate it explicitly.
For the case g > 0 the tau-function is an analytic torsion [3], and for the
case g=0 - the regularized energy of a certain natural pseudo-measure on the
interior domain of a closed curve. For these cases we rigorously prove the Ward
identities for the current correlation functions and determine them explicitly.
For the case g > 0 these functions coincide with those obtained in [21,36]
using bosonization. For the case g=0 the tau-function we have defined coincides
with the tau-function introduced in [29,44,24] as a dispersionless limit of the
Sato's tau-function for the two-dimensional Toda hierarchy. As a corollary of
the Ward identities, we obtain recent results [44,24] on relations between
conformal maps of exterior domains and tau-functions. For this case we also
define a Hermitian metric on the space of all contours of given area. As
another corollary of the Ward identities we prove that the introduced metric is
Kahler and the logarithm of the tau-function is its Kahler potential.
