# Aspects of Holonomic quantum fields

Isomonodromic deformation and ising model

Part IV Two-Dimensional Models and Related Developments

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## Keywords

Free Fermion Linear Ordinary Differential Equation Deformation Equation Regular Singularity Irregular Singularity
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## References

## The general framework of monodromy preserving deformation with irregular singularities was given by

- [1][1] K. Ueno, Master's thesis, Kyoto University (1979).Google Scholar
- [2]See also B. Klares, Sur une classe de connexions relatives, C. R. Acad. Sc. Paris, 288 (1979), 205–208.Google Scholar
- [3]H. Flaschka and A. C. Newell, Monodromy and spectral preserving deformatoins I, preprint Clarkson college of tech. (1979).Google Scholar

## The proof of complete integrability and the general definition of T function was given by

- [4]M. Jimbo, T. Miwa and K. Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients I, preprint RIMS 315, Kyoto Univ. (1980).Google Scholar

## As for Schlesinger transformations and T-quotients, see

- [5]M. Jimbo and T. Miwa, Monodromy preserving deformation of linear ordinary differential euqations with rational coefficients II, preprint RIMS 316, Kyoto Univ. (1980).Google Scholar

## In this chapter, we have not mentioned special examples such as Schlesinger equation and Painlevé equations. Schlesinger equation is discussed in chapter 1. As for Painlevé equations, see

- [6]K. Okamoto, Polynomial Hamiltonians associated to the Painlevé equations, preprint Tokyo Univ. (1980).Google Scholar

## References There are an enormous number of literatures concerning the Ising model. see e.g.

- [1]B.M. McCoy and T.T. Wu, The two-dimensional Ising model, Harvard University Press, 1973.Google Scholar

## Explicit formulas of the spin operators and correlation functions on the lattice are given from the viewpoint of the Clifford group in

- [2]M. Sato, T. Miwa and M. Jimbo, Holonomic quantum fields V, preprint RIMS 267, Kyoto Univ. (1978).Google Scholar

## Some recent developments in this branch are quoted in the references of [2]. Deformation theory and closed expression of correlation functions were introduced by

- [3]M. Sato, T. Miwa and M. Jimbo, Proc. Japan Acad. 53A (1977), 147–152, 153-158, 183-185.Google Scholar

## This work is a generalization of the monumental papers

- [4]E. Barouch, B.M. McCoy and T.T. Wu, Phys. Rev. Lett. 31 (1973) 1409–1411, T.T. Wu, B.M. McCoy, C.A.,Tracy and E. Barouch, Phys. Rev. B13 (1976) 316-374. in which the 2 point functions are obtained in terms of a Painlevé transcendent of the third kind. Details along with some generalizations of [3] are found inCrossRefGoogle Scholar
- [5]M. Sato, T. Miwa and M. Jimbo, Publ. RIMS, Kyoto Univ. 15 (1979) 577–629. or[[5a] —, Holonomic quantum fields IV, to appear in Publ. RIMS, Kyoto Univ. 15 (1980).Google Scholar
- [5b]—, Supplement to Holonomic quantum fields IV, preprint RIMS 304, Kyoto Univ. (1979).Google Scholar

## Extension of the above sheme to higher dimensions is briefly touched in

- [6]M. Jimbo, T. Miwa, M. Sato and Y. Môri, Holonomic quantum fields-The unanticipated link between deformation theory of differential equations and quantum fields-, preprint RIMS 305, Kyoto Univ. (1979) and the references cited therein.Google Scholar

## References This review is based on

- [1]M. Sato, T. Miwa and M. Jimbo, Holonomic quantum fields V, RIMS preprint 267 (1978), to appear in Publ. RIMS, Kyoto Univ.Google Scholar
- [2]M. Jimbo, T. Miwa and M. Sato, Studies on holonomic quantum fields XV, Proc. Japan Acad. 55 A (1979) 267–272.Google Scholar
- [3]M. Jimbo, T. Miwa, Y. Mori and M. Sato, Density matrix of impenetrable bose gas and the fifth Painlevé transcendent, RIMS preprint 303 (1979), to appear in Physics D.Google Scholar

## In [2] the correlation function for the non-linear Schrödinger equation with finite particle density at the infinite coupling is studied in detail. See also

- [4]H. G. Vaidya and C. A. Tracy, One particle reduced density matrix of impenetrable bosons in one dimension at zero temperature, J. Math. Phys. 20 (1979), 2291–2312.CrossRefGoogle Scholar

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© Springer-Verlag 1980