Abstract
A method of studing of double scaling limits for the two-dimensional string models of quantum gravity is stated. It is actually shown that the study of such limits reduces to the isomonodromic deformation method for integrable discrete equations. A relationship is indicated between the “universality” and isomonodromy properties of the model. It is shown that the partition function of the model is the τ-function associated with the fourth Painlevé equation (ℙ4), and also the Volterra chain. We consider in detail the properties of the Bäcklund transformations for ℙ4. Bibliography: 19 titles.
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Literature Cited
E. Brezin and V. Kazakov,Phys. Lett. B.,236B, 144 (1990).
D. Gross and A. Migdal,Phys. Rev. Lett.,64, 127 (1990).
M. Douglas and S. Shenker, “Strings in less than one dimension,” (Rutgers preprint), No. RU-89-34 (1989).
D. Gross and A. Migdal, “A nonperturbative treatment of two dimensional quantum gravity,” (Princeton preprint), No. PUPT-1159 (1989).
M. Douglas, “Strings in less than one dimension and the generalized KdV hierarchies,” (Rutgers preprint), No. RU-89-51 (1989).
T. Banks, M. Douglas, N. Seiberg, and S. Shenker, “Microscopic and macroscopic loops in non-perturbative two dimensional gravity,” (Rutgers preprint), No. RU-89-50 (1989).
O. Alvarez and P. Windey, “Universality in two dimensional quantum gravity,” (Preprint), Paris VI University, No. PAR-LPTHE 90-12 (1990).
E. Brezin, E. Marinari, and G. Parisi, “A non-perturbative ambiguity free solution of a string model,” (Roma preprint), No. ROM2F-90-09 (1990).
M. Douglas, N. Seiberg, and S. Shenker, “Flow and instability in quantum gravity,” (Rutgers preprint), No. RU-90-19 (1990).
G. Moore, “Geometry of the string equations” (Preprint), Yale University, No. YCTP-P4-90 (1990).
H. Bateman and A. Erdélyi,Higher Transcendental Functions, Mc Graw-Hill, New York-Toronto-London (1953).
A. R. It-s,Izv. Akad. Nauk SSSR, Ser. Mat.,49, No. 3, 530–565 (1985).
A. R. It-s and V. Yu. Novokshenov, “The isomonodromic deformation method in the theory of Painlevé equations,”Lecture Notes in Math.,1191, Springer-Verlag (1986).
A. R. It-s, A. A. Kapaev, and A. V. Kitaev, in:Problems of Theoretical Physics, III. Elementary Particle Theory. Quantum Mechanics. Mathematical Physics. Statistical Physics [in Russian], LGU, Leningrad, 182–192 (1988).
A. V. Kitaev,Teor. Mat. Fiz.,64, No. 3, 347–369 (1985).
V. I. Gromak,Differents. Uravn.,14, No. 12, 2131–2135 (1978).
A. V. Kitaev,Zapiski Nauch. Sem. LOMI,169, 84–89 (1988).
M. Kac and P. van Moerbeke,Advances in Math.,16, No. 2, 160–169 (1975).
M. Jimbo and T. Miwa,Physica D.,2D, No. 3, 407–448 (1981).
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Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 187, pp. 3–30, 1990.
Translated by O. A. Ivanov.
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It-s, A.R., Kitaev, A.V. & Fokas, A.S. Matrix models of two-dimensional quantum gravity and isomonodromic solutions of “discrete Painlevé” equations. J Math Sci 73, 415–429 (1995). https://doi.org/10.1007/BF02364564
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DOI: https://doi.org/10.1007/BF02364564