Monodromy fields on \({\mathbb{Z}}^ 2\)
This paper concerns lattice field theory analogues of the two dimensional Ising model. The field operators are well defined elements of an infinite dimensional spin group and may be studied in terms of their induced rotations. One of the principal results is a diagonalization of the matrix elements of the induced rotation of a single field operator relative to the splitting of the Hilbert space which determines the infinite volume spin representation. This diagonalization has an intimate relation with the function theory on an elliptic curve (the dispersion curve for the induced rotation of the transfer matrix) which, although explicit, is not well understood. The second part of the paper concerns lattice analogues of the wave functions introduced by Sato, Miwa, and Jimbo in their analysis of monodromy preserving deformations of the Euclidean Dirac operator. I now believe the point of view in this section is wrong headed. What is actually constructed in this section is clearly a Green function for a lattice version of the Euclidean Dirac equation. The associated lattice Dirac operator ought to be thought of as acting on the sections of ''bundle'' which incorporates the appropriate monodromy. The tau functions introduced by Sato, Miwa and Jimbo should turn out to be regularized determinants for the two dimensional Dirac operator. In the continuum limit the same remarks apply. I believe the whole theory will be clarified once the Green function for the two dimensional Dirac operator is taken as the fundamental object and the wave functions introduced by Sato, Miwa and Jimbo are understood to measure the asymptotic behavior of this Green function in the vicinity of the branch points. The third section of this paper proves the convergence of the continuum scaling limit in the massive regime. This proof may be simplified using some elliptic function representations for the dispersion curve of the induced rotation for the transfer matrix. It also turns out that the use of the rather heavy product deformation theory from an earlier paper may be completely avoided if one just takes the group theory described earlier in the paper more seriously.
scientific article; zbMATH DE number 3994151
scientific article