Abstract
In this paper we define tau functions for holonomic fields associated with the Dirac operator on the Poincaré disk. The deformation analysis of the tau functions is worked out and in the case of the two point function, the tau function is expressed in terms of a Painlevé function of type VI.
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References
Barouch, E., McCoy, B.M., Wu, T.T.: Zero-field suceptibility of the two-dimensional Ising model nearT c . Phys. Rev. Lett.31, 1409–1411 (1973)
Fay, J.D.: Fourier coefficients for the resolvent for a Fuchsian group. J. Reine Angew. Math.294, 143–203 (1977)
Hartman, P.: Ordinary Differential Equations. New York: John Wiley & Sons, 1964
Ince, E.L.: Ordinary Differential Equations. New York: Dover, 1947
Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformation theory of differential equations with rational coefficients. I. General theory and the τ function. Physica2D, 306–352 (1981)
Narayanan, R., Tracy, C.A.: Holonomic quantum field theory of Bosons in the Poincaré disk and the zero curvature limit. Nucl. Phys.B340, 568–594 (1990)
Okamoto, K.: On the τ-function of the Painlevé equations. Physica2D, 525–535 (1981), Studies on the Painlevé equations. I. Sixth PainlevéP VI. Annali di Mathematica Pura and Applicata, vol.CXLVI, 337–381 (1987)
McCoy, B.M., Tracy, C.A., Wu, T.T.: Painlevé functions of the third kind. J. Math. Phys.18, 1058–1092 (1977)
Palmer, J., Tracy, C.: Two dimensional Ising correlations: The SMJ analysis. Adv. in Appl. Math.4, 46–102 (1983)
Palmer, J.: Monodromy fields on Z2. Commun. Math. Phys.102, 175–206 (1985)
Palmer, J.: Determinants of Cauchy-Riemann operators as τ functions. Acta Applicande Mathematicae18, No. 3, 199–223 (1990)
Palmer, J.: Tau functions for the Dirac operator in the Euclidean plane. Pac. Math.160, 259–342 (1993)
Palmer, J., Tracy, C.A.: Monodromy preserving deformations of the Dirac operator on the hyperbolic plane. In: Mathematics of Nonlinear Science: Proceedings of an AMS special session held January 11–14, 1989, M.S. Berger (ed.) Contemp. Math.108, 119–131 (1989)
Pressley, A., Segal, G.: Loop Groups. Oxford: Clarendon Press, 1986
Sato, M., Miwa, T., Jimbo, M.: Holonomic Quantum Fields I–V Publ. RIMS, Kyoto Univ.14, 223–267 (1978),15, 201–278 (1979),15, 577–629 (1979),15, 871–972 (1979)16, 531–584 (1980)
Segal, G., Wilson, G.: Loop groups and equations of KdV type. Pub. Math. I.H.E.S.61, 5–65 (1980)
Tracy, C.A., McCoy, B.M.: Neutron scattering and the correlations of the Ising model nearT c. Phys. Rev. Lett.31, 1500–1504 (1973)
Tracy, C.A.: Monodromy preserving deformation theory for the Klein-Gordon equation in the hyperoblic plane. Physica34D, 347–365 (1989)
Tracy, C.A.: Monodromy preserving deformations of linear ordinary and partial differential equations. In: Solutions in Physics, Mathematics, and Nonlinear Optics, Oliver, P.J. and Sattinger D.H. (eds.) Berlin Heidelberg New York: Springer 1990, pp. 165–174
Wu, T.T., McCoy, B.M., Tracy, C.A., Barouch, E.: Spin-spin correlation functions for the two dimensional Ising model: Exact theory in the scaling region. Phys. Rev.B 13, 316–374 (1976)
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Communicated by M. Jimbo
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Palmer, J., Beatty, M. & Tracy, C.A. Tau functions for the Dirac operator on the Poincaré disk. Commun.Math. Phys. 165, 97–173 (1994). https://doi.org/10.1007/BF02099740
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DOI: https://doi.org/10.1007/BF02099740