Free bosons and taufunctions for compact Riemann surfaces and closed smooth Jordan curves I. Current correlation functions
Abstract
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 g1, and for the case g=0  by smooth closed Jordan curves on the complex plane. In both cases we define a notion of taufunction as a partition function of the theory and evaluate it explicitly. For the case g > 0 the taufunction is an analytic torsion [3], and for the case g=0  the regularized energy of a certain natural pseudomeasure 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 taufunction we have defined coincides with the taufunction introduced in [29,44,24] as a dispersionless limit of the Sato's taufunction for the twodimensional Toda hierarchy. As a corollary of the Ward identities, we obtain recent results [44,24] on relations between conformal maps of exterior domains and taufunctions. 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 taufunction is its Kahler potential.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 2001
 arXiv:
 arXiv:math/0102164
 Bibcode:
 2001math......2164T
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Complex Variables;
 High Energy Physics  Theory;
 14H10;
 30C35;
 81T40
 EPrint:
 48 pages