Monodromy fields on \({\mathbb{Z}}^ 2\)
type: Item
instance of: scholarly article
title: Monodromy fields on \({\mathbb{Z}}^ 2\)
zbMATH Open document ID: 0614.35079
DOI: 10.1007/BF01229376
author: John Palmer
published in: Communications in Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 81T08
zbMATH DE Number: 3994151
zbMATH Keywords: lattice field theory
zbMATH Keywords: Ising model
zbMATH Keywords: spin group
zbMATH Keywords: infinite volume spin representation
zbMATH Keywords: diagonalization
zbMATH Keywords: wave functions
zbMATH Keywords: monodromy preserving deformations
zbMATH Keywords: Green function
zbMATH Keywords: Euclidean Dirac equation
zbMATH Keywords: convergence
zbMATH Keywords: continuum scaling limit
zbMATH Keywords: massive regime
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01229376
OpenAlex ID: W1511967064
description: scientific article; zbMATH DE number 3994151