Holonomic quantum field theory of bosons in the Poincare disk and the zero curvature limit

We formulate a holonomic quantum field theory of bosons on a Poincare disk of radius R (DR) based on a monodromy preserving deformation of the laplacian operator defined on DR. First the isomonodromy problem for the laplacian is converted to an isomonodromy problem for a fuchsian system using the hyperbolic Laplace transform defined on DR. The deformation equations for the fuchsian system (Schlesinger equations) and an associated closed one-form ω are then discussed. Locally ω = d log τ defines a τ-function which is then identified with an n-point function of bosons defined on DR. At every step we discuss the limit R → ∞ where the problem reduces to one on the euclidean plane. This facilitates a detailed comparison with the original results of Sato, Miwa and Jimbo on the euclidean plane. The two-point function is discussed in some detail and it is shown that it can be expressed in terms of a Painleve transcendent of the sixth kind. (orig.)