SMJ analysis of monodromy fields.

Abstract

The connection discovered by M. Sato, T. Miwa and M. Jimbo (SMJ) between the monodromy-preserving deformation theory of the two-dimensional Euclidean Dirac operator and quantum fields is rigorously established for the case of nonreal S¹ monodromy parameters. This connection involves the expression of the associated n-point functions in terms of solutions to deformation equations which arise as necessary conditions for the monodromy exhibited by a class of multivalued solutions of the Euclidean Dirac equation to be preserved under perturbations of branch points. Our approach utilizes recent results involving infinite-dimensional group representations. A lattice version of the n-point function is introduced as a section of a determinant bundle defined over an infinite dimensional Grassmannian. A trivialization for this bundle is singled out so that the corresponding n-point functions behave like Ising correlations in the massive scaling regime. Then the SMJ n-point functions are recovered as the scaled functions. A parallel scaling analysis is carried out with lattice analogues of the Euclidean Dirac wave functions which scale to square-integrable multivalued solutions of the Euclidean Dirac equation and the connection between the SMJ deformation theory and the n-point functions is rigorously established in terms of local Fourier expansion coefficients of these wave functions. These results are presented in detail for two-point functions with the same monodromy associated to each site.