9401594 Palmer Holonomic fields are quantum field theories with an intimate connection to elliptic operators with isolated singularities. The correlation function or tau-function of such fields have appeared in a wide variety of contexts including, the scaling limit of the two dimensional Ising model, the Riemann-Hilbert problem on the complex sphere, level spacing distributions for random matrices, the KdV hierarchy and its relatives, the asymptotics of Toeplitz determinants, and recently intersection theory on the moduli space of Riemann surfaces. A reformulation of the original Sato, Miwa and Jimbo theory that has been worked out by Professor Palmer makes it possible to consider a variety of generalizations. The proposal here is to complete the analysis of the Dirac operators on the hyperbolic disk and to make the field theory connection by developing an analytic version of the theory of vertex operators. These vertex operators should also prove useful in simplifying (and generalizing) some of the algebraic analysis that goes into the theory of vertex operator algebras. Quantum field theories are mathematical models for quantum mechanics. In addition the use in mathematical physics in these theories often represent the beginning of new developments in mathematics. The proposal here involves explicit description of invariants of earlier quantum field theories. These descriptions lead to surprising connections with important recent mathematical developments in algebra and geometry. ***
