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

摘要

We study families of quantum field theories of free bosons on a compactRiemann surface of genus g. For the case g > 0 these theories are parameterizedby holomorphic line bundles of degree g-1, and for the case g=0 - by smoothclosed Jordan curves on the complex plane. In both cases we define a notion oftau-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 thecase g=0 - the regularized energy of a certain natural pseudo-measure on theinterior domain of a closed curve. For these cases we rigorously prove the Wardidentities 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 coincideswith the tau-function introduced in [29,44,24] as a dispersionless limit of theSato's tau-function for the two-dimensional Toda hierarchy. As a corollary ofthe Ward identities, we obtain recent results [44,24] on relations betweenconformal maps of exterior domains and tau-functions. For this case we alsodefine a Hermitian metric on the space of all contours of given area. Asanother corollary of the Ward identities we prove that the introduced metric isKahler and the logarithm of the tau-function is its Kahler potential.