Free bosons and tau-functions for compact Riemann surfaces and closed smooth Jordan curves. Current correlation functions

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 τ-function as a partition function of the theory and evaluate it explicitly. For the case g > 0 the τ-function is an analytic torsion, 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 by using bosonization. For the case g = 0, the τ-function we have defined coincides with the τ-function introduced as a dispersionless limit of the Sato's τ-function for the two-dimensional Toda hierarchy. As a corollary of the Ward identities, we note some recent results on relations between conformal maps of exterior domains and τ-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 Kähler and the logarithm of the τ-function is its Kähler potential.