Geometry of the string equations. (English) Zbl 0727.35134

This paper presents an attempt to construct a mathematical framework for the string equations of Hermitian and unitary 2D matrix models. The physical parameters defining the string equations are interpreted as moduli of meromorphic gauge fields and the compatibility conditions can be interpreted as defining a “quantum” Riemann space. As an application of this formalism some properties of the Brezin-Marinari-Parisi solutions [see E. Brezin, E. Marinari and G. Parisi, A non- perturbative ambiguity free solution of a string model, ROM 2F 90-09] are discussed and a connection to twistor theory is briefly mentioned.


35Q75 PDEs in connection with relativity and gravitational theory
81R25 Spinor and twistor methods applied to problems in quantum theory
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI


[1] Brézin, E., Kazakov, V.: Exactly solvable field theories of closed strings. Phys. Lett.B236, 144 (1990)
[2] Douglas, M., Shenker, S.: String in less than one dimension. Rutgers preprint RU-89-34
[3] Gross, D., Migdal, A.: Nonperturbative two dimensional quantum gravity. Phys. Rev. Lett.64, 127 (1990) · Zbl 1050.81610
[4] Gross, D., Migdal, A.: A nonperturbative treatment of two-dimensional quantum gravity. Princeton preprint PUPT-1159 (1989)
[5] Douglas, M.: Strings in less one dimension and the generalized KdV hierarchies. Rutgers preprint RU-89-51
[6] Banks, T., Douglas, M., Seiberg, N., Shenker, S.: Microscopic and macroscopic loops in non-perturbative two dimensional gravity. Rutger preprint RU-89-50 · Zbl 1332.81200
[7] Witten, E.: On the structure of the topological phase of two dimensional gravity. Preprint IASSNS-HEP-89/66
[8] Distler, J.: 2D quantum gravity, topological field theory and multicritical matrix models. Princeton preprint PUPT-1161
[9] Dijkgraaf, R., Witten, E.: Mean field theory, topological field theory, and multimatrix models. IASSNS-HEP-90/18; PUPT-1166
[10] Verlinde, E., Verlinde, H.: A solution of two dimensional topological quantum gravity. Preprint IASSNS-HEP-90/40 · Zbl 0990.81761
[11] Friedan, D., Shenker, S.: The analytic geometry of two-dimensional conformal field theory. Nucl. Phys. B281, 509 (1987); Friedan, D.: A new formulation of string theory. Physica Scripta T15, 72 (1987) · Zbl 0559.58010
[12] Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformations groups for soliton equations. I. Proc. Jpn. Acad.57A, 342 (1981); Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformations groups for soliton equations. II. Ibid. Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformations groups for soliton equations I. Proc. Jpn. Acad.57A, 387; III. J. Phys. Soc. Jpn.50, 3806 (1981); IV. Physica4D, 343 (1982); V. Publ. RIMS, Kyoto University18, 1111 (1982); VI. J. Phys. Soc. Jpn.50, 3813 (1981); VII. Publ. RIMS, Kyoto University18, 1077 (1982) · Zbl 0571.35102
[13] Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. I.H.E.S.61, 1 (1985) · Zbl 0592.35112
[14] Frenkel, I.: Representations of affine Lie algebras, hecke modular forms, and Korteweg-de Vries type equations. Proceedings of the 1981 Rutgers Conference on Lie Algebras and related topics. Lecture Notes in Mathematics, vol. 933, p. 71. Berlin, Heidelberg, New York: Springer 1982 · Zbl 0505.17008
[15] Flaschka, H., Newell, A.: Monodromy and spectrum-preserving deformations. I. Commun. Math. Phys.76, 65 (1980) · Zbl 0439.34005
[16] Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: Differential equations for quantum correlation functions. Australian National University preprint; Trieste preprints, IC/89/120,107,139 · Zbl 0719.35091
[17] Martinec, E.: Private communication
[18] Drinfeld, Sokolov: Equations of Korteweg-de Vries type and simple Lie algebras. Sov. J. Math. 1975 (1985) · Zbl 0578.58040
[19] Gelfand, I.M., Dickii, L.A.: Asymptotic behavior of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-De Vries equations. Russian Math. Surv.30, 77 (1975) · Zbl 0334.58007
[20] Periwal, V., Shevitz, D.: Unitary-matrix models as exactly solvable string theories. Phys. Rev. Lett.64, 1326 (1990)
[21] Crnkovic, C., Douglas, M., Moore, G.: To appear
[22] Kutasov, D., Di Francesco. Ph.: Unitary minimal models coupled to 2D quantum gravity. Princeton preprint, PUPT-1173
[23] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351 (1989) · Zbl 0667.57005
[24] Ginsparg, P., Goulian, M., Plesser, M.R., Zinn-Justin, J.: (p,q) string actions. Harvard preprint HUTP-90/A015;SPhT/90-049
[25] Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with radional coefficients. Physica2D, 306 (1981) · Zbl 1194.34167
[26] Jimbo, M., Miwa, T.: Monodromy preserving deformation of linear ordinary differential equations with radional coefficients. II. Physica2D, 407 (1981) · Zbl 1194.34166
[27] Sato, M., Miwa, T., Jimbo, M.: Aspects of holonomic quantum fields isomonodromic deformation and Ising model. In: Complex Analysis, Microlocal Calculus and Relativistic Quantum Theorey. Iagolnitzer, D. (ed.). Lecture Notes in Physics, vol. 126. Berlin, Heidelberg, New York: Springer · Zbl 0451.34008
[28] Jimbo, M.: Introduction to holonomic quantum fields for mathematicians. Proc. Symp. Pure Math.49, part I. 379 (1989) · Zbl 0676.53087
[29] Its, A., Novokshenov, V.Yu.: The isomonodromic deformation method in the theory of Painlevé equations. Lecture Notes Mathematics, vol. 1191. Berlin, Heidelberg, New York: Springer · Zbl 0592.34001
[30] Kapaev, A.: Asymptotics of solutions of the Painlevé equation of the first kind. Differential Equations24, 1107 (1988) · Zbl 0677.34052
[31] Miwa, T.: Painlevé property of monodromy preserving deformation equations and the analytic of {\(\tau\)} functions. Publ. Res. Inst. Math. Sci.17, 703 (1981) · Zbl 0605.34005
[32] Brezin, E., Marinari, E., Parisi, G.: A non-perturbative ambiguity free solution of a string model. Preprint ROM2F-90-09
[33] David, F.: Loop equations and non-perturbative effects in two-dimensional quantum gravity. Preprint SPhT/90-043
[34] Its, A.R., Novokshenov, V.Yu.: Effective sufficient conditions for the solvability of the inverse problem of monodromy theory for systems of linear ordinary differential equations. Funct. Anal. Appl.22, 190 (1988) · Zbl 0810.34012
[35] Wasow, W.: Asymptotic expansions for ordinary differential equations. New York: Interscience 1965 · Zbl 0133.35301
[36] Sibuya, Y.: Stokes phenomena. Bull. Am. Math. Soc.83, 1075 (1977) · Zbl 0386.34008
[37] Malgrange, B.: La classification des connexions irréguliers á une variable. In: Mathématique et Physique: Sém École Norm. Sup. 1979–1982. Basel: Birkhäuser 1983
[38] Jurkat, W.: Meromorphe Differentialgleichungen. Lecture Notes in Mathematics, vol. 637. Berlin, Heidelberg, New York: Springer · Zbl 0408.34004
[39] Babbitt, D., Varadarajan, V.: Local moduli for meromorphic differential equations. Bull. Am. Math. Soc.72, 95 (1985) · Zbl 0579.34005
[40] Babitt, D., Varadarajan, V.: Deformations of nilpotent matrices over rings and reduction of analytic families of meromorphic differential equations. Mem. Am. Math. Soc.55, 1 (1985)
[41] Varadarajan, V.: Recent progress in differential equations in the complex domain. Preprint
[42] Majima, J.: Asymptotic analysis for integrable connections with irregular singular points. Lecture Notes in Mathematics, vol. 1075. Berlin, Heidelberg, New York: Springer · Zbl 0546.58003
[43] Dubrovin, Matveev, Novikov: Non-linear equations of Korteweg-De Vries type, finite-zone linear operators, and abelian varieties. Russ. Math. Surveys31, 59 (1976) · Zbl 0346.35025
[44] Mumford, D.: An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de Vries equation and related non-linear equations. Proceedings of symposium on algebraic geometry. Nagata, M. (ed.) Kinokuniya, Tokyo, 1978 · Zbl 0423.14007
[45] Abramowitz, M., Stegun, J.: Handbook of mathematical functions. Dover · Zbl 0171.38503
[46] Sato, M., Miwa, T., Jimbo, M.: Holonomic quantum field theory. II. Publ. RIMS15, 201 (1979) · Zbl 0433.35058
[47] Miwa, T.: Clifford operators and Riemann’s monodromy problem. Publ. Res. Inst. Math. Sci.17, 665 (1981) · Zbl 0505.35070
[48] Zamalodchikov, Al.B.: Conformal scalar field on the hyperelliptic curve and critical Ashkin-Teller multipoint correlation functions. Nucl. Phys. B285, 481 (1987)
[49] Bershadsky, M., Radul, A.: Conformal field theories with additionalZ N symmetry. Int. J. Mod. Phys.A2, 165 (1987) · Zbl 1165.81373
[50] Dixon, L., Friedan, D., Martinec, E., Shenker, S.: The conformal field theory of Orbifolds. Nucl. Phys. B282, 13 (1987)
[51] Hamidi, S., Vafa, C.: Interactions on Orbifolds. Nucl. Phys. B279, 465 (1987)
[52] Barouch, E., McCoy, B.M., Wu, T.T.: Phys. Rev. Lett.31, 1409 (1973); Wu, T.T., McCoy, B.M., Tracy, C.A., Barouch, E.: Phys. Rev. B13, 316 (1976)
[53] Witten, E.: Conformal field theory, Grassmanians, and algebraic curves. Commun. Math. Phys.113, 189 (1988) · Zbl 0636.22012
[54] Palmer, J.: Determinants of Cauchy-Riemann operators as {\(\tau\)}-functions. Univ. of Arizona preprint; The tau function for Cauchy-Riemann operators onS 2. Unpublished letter to C. Tracey
[55] Mehta, M.L.: Random matrices. New York: Academic Press 1967 · Zbl 0925.60011
[56] See Sect. 10.3 in Itzykson, C. and Drouffe, J.-M.: Statistical field theory, vol. 2. Cambridge: Cambridge Univ. Press 1989
[57] Jimbo, M., Miwa, T., Mori, Y., Sato, M.: Density matrix of an impenetrable bose gas and the fifth Painlevé transcendent. Physica1D, 80 (1980) · Zbl 1194.82007
[58] Morozov, A., Shatashvili, S.: Private communications, Nov. 1989, and Dec. 1989
[59] Moore, G., Seiberg, N.: Lectures of RCFT. Preprint RU-89-32; YCTP-P13-89
[60] Brezin, E., Marinari, E., Parisi, G.: A non-perturbative ambiguity free solution of a string model. ROM2F-90-09
[61] Douglas, M., Seiberg, N., Shenker, S.: Flow and instability in quantum gravity. Rutgers preprint, RU-90-19
[62] See reference [29], Its, A., Novokshenov, V.Yu.: The isomonodromic deformation method in the theory of Painlevé equations. Lecture Notes Mathematics, vol. 1191. Berlin, Heidelberg, New York: Springer especially, Chap. 5
[63] Crnkovic, C., Ginsparg, P., Moore, G.: The Ising model, the Yang-Lee edge singularity, and 2D quantum gravity. Phys. Lett.237B, 196 (1990)
[64] Hastings, S.P., McLeod, J.B.: A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation. Arch. Rat. Mech. Anal.73, 31 (1980) · Zbl 0426.34019
[65] Mason, Sparling: Nonlinear Schrödinger and Korteweg-De Vries are reductions of self-dual Yang-Mills. Phys. Lett.137A, 29 (1989)
[66] Belavin, A., Zakharov, V.: Yang-Mills equations as inverse scattering problem. Phys. Lett.73B, 53 (1978)
[67] Atiyah, M.F.: Collected works, vol. 5. Oxford: Clarendon press 1988 · Zbl 0691.53003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.