Abstract
The string equations of hermitian and unitary matrix models of 2D gravity are flatness conditions. These flatness conditions may be interpreted as the consistency conditions for isomonodromic deformation of an equation with an irregular singularity. In particular, the partition function of the matrix model is shown to be the tau function for isomonodromic deformation. The physical parameters defining the string equation are interpreted as moduli of meromorphic gauge fields, and the compatibility conditions can be interpreted as defining a “quantum” analog of a Riemann surface. In the latter interpretation, the equations may be viewed as compatibility conditions for transport on “quantum moduli space” of correlation functions in a theory of free fermions. We discuss how the free fermion field theory may be deduced directly from the matrix model integral. As an application of our formalism we discuss some properties of the BMP solutions of the string equations. We also mention briefly a possible connection to twistor theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Brézin, E., Kazakov, V.: Exactly solvable field theories of closed strings. Phys. Lett.B236, 144 (1990)
Douglas, M., Shenker, S.: String in less than one dimension. Rutgers preprint RU-89-34
Gross, D., Migdal, A.: Nonperturbative two dimensional quantum gravity. Phys. Rev. Lett.64, 127 (1990)
Gross, D., Migdal, A.: A nonperturbative treatment of two-dimensional quantum gravity. Princeton preprint PUPT-1159 (1989)
Douglas, M.: Strings in less one dimension and the generalized KdV hierarchies. Rutgers preprint RU-89-51
Banks, T., Douglas, M., Seiberg, N., Shenker, S.: Microscopic and macroscopic loops in non-perturbative two dimensional gravity. Rutger preprint RU-89-50
Witten, E.: On the structure of the topological phase of two dimensional gravity. Preprint IASSNS-HEP-89/66
Distler, J.: 2D quantum gravity, topological field theory and multicritical matrix models. Princeton preprint PUPT-1161
Dijkgraaf, R., Witten, E.: Mean field theory, topological field theory, and multimatrix models. IASSNS-HEP-90/18; PUPT-1166
Verlinde, E., Verlinde, H.: A solution of two dimensional topological quantum gravity. Preprint IASSNS-HEP-90/40
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)
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)
Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. I.H.E.S.61, 1 (1985)
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
Flaschka, H., Newell, A.: Monodromy and spectrum-preserving deformations. I. Commun. Math. Phys.76, 65 (1980)
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
Martinec, E.: Private communication
Drinfeld, Sokolov: Equations of Korteweg-de Vries type and simple Lie algebras. Sov. J. Math. 1975 (1985)
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)
Periwal, V., Shevitz, D.: Unitary-matrix models as exactly solvable string theories. Phys. Rev. Lett.64, 1326 (1990)
Crnkovic, C., Douglas, M., Moore, G.: To appear
Kutasov, D., Di Francesco. Ph.: Unitary minimal models coupled to 2D quantum gravity. Princeton preprint, PUPT-1173
Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351 (1989)
Ginsparg, P., Goulian, M., Plesser, M.R., Zinn-Justin, J.: (p,q) string actions. Harvard preprint HUTP-90/A015;SPhT/90-049
Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with radional coefficients. Physica2D, 306 (1981)
Jimbo, M., Miwa, T.: Monodromy preserving deformation of linear ordinary differential equations with radional coefficients. II. Physica2D, 407 (1981)
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
Jimbo, M.: Introduction to holonomic quantum fields for mathematicians. Proc. Symp. Pure Math.49, part I. 379 (1989)
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
Kapaev, A.: Asymptotics of solutions of the Painlevé equation of the first kind. Differential Equations24, 1107 (1988)
Miwa, T.: Painlevé property of monodromy preserving deformation equations and the analytic of τ functions. Publ. Res. Inst. Math. Sci.17, 703 (1981)
Brezin, E., Marinari, E., Parisi, G.: A non-perturbative ambiguity free solution of a string model. Preprint ROM2F-90-09
David, F.: Loop equations and non-perturbative effects in two-dimensional quantum gravity. Preprint SPhT/90-043
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)
Wasow, W.: Asymptotic expansions for ordinary differential equations. New York: Interscience 1965
Sibuya, Y.: Stokes phenomena. Bull. Am. Math. Soc.83, 1075 (1977)
Malgrange, B.: La classification des connexions irréguliers á une variable. In: Mathématique et Physique: Sém École Norm. Sup. 1979–1982. Basel: Birkhäuser 1983
Jurkat, W.: Meromorphe Differentialgleichungen. Lecture Notes in Mathematics, vol. 637. Berlin, Heidelberg, New York: Springer
Babbitt, D., Varadarajan, V.: Local moduli for meromorphic differential equations. Bull. Am. Math. Soc.72, 95 (1985)
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)
Varadarajan, V.: Recent progress in differential equations in the complex domain. Preprint
Majima, J.: Asymptotic analysis for integrable connections with irregular singular points. Lecture Notes in Mathematics, vol. 1075. Berlin, Heidelberg, New York: Springer
Dubrovin, Matveev, Novikov: Non-linear equations of Korteweg-De Vries type, finite-zone linear operators, and abelian varieties. Russ. Math. Surveys31, 59 (1976)
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
Abramowitz, M., Stegun, J.: Handbook of mathematical functions. Dover
Sato, M., Miwa, T., Jimbo, M.: Holonomic quantum field theory. II. Publ. RIMS15, 201 (1979)
Miwa, T.: Clifford operators and Riemann's monodromy problem. Publ. Res. Inst. Math. Sci.17, 665 (1981)
Zamalodchikov, Al.B.: Conformal scalar field on the hyperelliptic curve and critical Ashkin-Teller multipoint correlation functions. Nucl. Phys. B285, 481 (1987)
Bershadsky, M., Radul, A.: Conformal field theories with additionalZ N symmetry. Int. J. Mod. Phys.A2, 165 (1987)
Dixon, L., Friedan, D., Martinec, E., Shenker, S.: The conformal field theory of Orbifolds. Nucl. Phys. B282, 13 (1987)
Hamidi, S., Vafa, C.: Interactions on Orbifolds. Nucl. Phys. B279, 465 (1987)
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)
Witten, E.: Conformal field theory, Grassmanians, and algebraic curves. Commun. Math. Phys.113, 189 (1988)
Palmer, J.: Determinants of Cauchy-Riemann operators as τ-functions. Univ. of Arizona preprint; The tau function for Cauchy-Riemann operators onS 2. Unpublished letter to C. Tracey
Mehta, M.L.: Random matrices. New York: Academic Press 1967
See Sect. 10.3 in Itzykson, C. and Drouffe, J.-M.: Statistical field theory, vol. 2. Cambridge: Cambridge Univ. Press 1989
Jimbo, M., Miwa, T., Mori, Y., Sato, M.: Density matrix of an impenetrable bose gas and the fifth Painlevé transcendent. Physica1D, 80 (1980)
Morozov, A., Shatashvili, S.: Private communications, Nov. 1989, and Dec. 1989
Moore, G., Seiberg, N.: Lectures of RCFT. Preprint RU-89-32; YCTP-P13-89
Brezin, E., Marinari, E., Parisi, G.: A non-perturbative ambiguity free solution of a string model. ROM2F-90-09
Douglas, M., Seiberg, N., Shenker, S.: Flow and instability in quantum gravity. Rutgers preprint, RU-90-19
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
Crnkovic, C., Ginsparg, P., Moore, G.: The Ising model, the Yang-Lee edge singularity, and 2D quantum gravity. Phys. Lett.237B, 196 (1990)
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)
Mason, Sparling: Nonlinear Schrödinger and Korteweg-De Vries are reductions of self-dual Yang-Mills. Phys. Lett.137A, 29 (1989)
Belavin, A., Zakharov, V.: Yang-Mills equations as inverse scattering problem. Phys. Lett.73B, 53 (1978)
Atiyah, M.F.: Collected works, vol. 5. Oxford: Clarendon press 1988
Author information
Authors and Affiliations
Additional information
Communicated by N. Yu. Reshetikhin
Rights and permissions
About this article
Cite this article
Moore, G. Geometry of the string equations. Commun.Math. Phys. 133, 261–304 (1990). https://doi.org/10.1007/BF02097368
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02097368