On the origin of integrability in matrix models. (English) Zbl 0732.35100
One-matrix model \(Z=\int {\mathcal D}\Phi e^{-V(\Phi)}\), \(V(\Phi)=\sum t_ ktr\{\Phi^ k\}\) (unitary one-matrix model, two-matrix model) of string theory is related to Toda hierarchy (quaternionic Toda hierarchy, 2-d Toda hierarchy) by using the orthogonal polynomial transcription of the matrix problem. Appropriately defined \(\tau\)-function (different from the standard free fermion setting) of the Toda lattice is realized by a matrix model partition function.
Reviewer: J.Chrastina (Brno)
MSC:
| 35Q75 | PDEs in connection with relativity and gravitational theory |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 83C27 | Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory |
| 15A90 | Applications of matrix theory to physics (MSC2000) |
References:
| [1] | Brezin, E., Kazakov, V.: Exactly solvable field theories of closed strings. Phys. Lett.236B, 144 (1990); Douglas, M., Shenker, S.: Strings in less than one dimension. Nucl. Phys.B335, 635 (1990) |
| [2] | Gross, D., Migdal, A.: Nonperturbative two-dimensional quantum gravity. Phys. Rev. Lett.64, 127 (1990) · Zbl 1050.81610 · doi:10.1103/PhysRevLett.64.127 |
| [3] | Banks, T., Douglas, M., Seiberg, N., Shenker, S.: Microscopic and macroscopic loops in nonperturbative 2d quantum gravity. Phys. Lett.238B, 279 (1990) · Zbl 1332.81200 |
| [4] | Douglas, M.: Strings in less than one dimension and the generalized KdV hierarchies. Phys. Lett.238B, 176 (1990) · Zbl 1332.81211 |
| [5] | Fukuma, M., Kawai, H., Nakayama, R.: Continuum Schwinger-Dyson equations and universal structures in 2d quantum gravity. Tokyo preprint UT-562 (May 1990); Dijkgraaf, R., Verlinde, E., Verlinde, H.: Loop equations and Virasoro constraints in nonperturbative 2d quantum gravity. Princeton preprint PUPT-1184 (May 1990) |
| [6] | Seiberg, N.: Notes on quantum Liouville theory and quantum gravity. Rutgers preprint RU-90-29 (June 1990) |
| [7] | Das, S. R., Jevicki, A.: String field theory and physical interpretation ofd=1 strings. Brown preprint HET-750 (1990) · Zbl 1020.81765 |
| [8] | Shenker, S.: The strength of nonperturbative effects in string theory. Rutgers preprint RU-90-47 (August 1990) |
| [9] | Witten, E.: Two-dimensional gravity and intersection theory on moduli space. IAS preprint IASSNS-HEP-90/45 (May 1990) |
| [10] | Gerasimov, A., Marshakov, A., Mironov, A., Morozov, A., Orlov, A.: Matrix models of 2d gravity and Toda theory. ITEP preprint (July 1990) |
| [11] | Bessis, D., Itzykson, C., Zuber, J.-B.: Quantum field theory techniques in graphical enumeration. Adv. Appl. Math.1, 109 (1980) · Zbl 0453.05035 · doi:10.1016/0196-8858(80)90008-1 |
| [12] | Moser, J.: Finitely many mass points on the line under the influence of an exponential potential–an integrable system. In: Lecture Notes in Physics, vol.38. Moser, J. (ed.) Berlin, Heidelberg, New York: Springer 1975 · Zbl 0323.70012 |
| [13] | Adler, M.: On a trace functional for formal pseudo-differential operators and the symplectic structure of the KdV equations. Invent. Math.50, 219 (1979) · Zbl 0393.35058 · doi:10.1007/BF01410079 |
| [14] | Kostant, B.: The solution to a generalized Toda lattice and representation theory. Adv. Math.34, 195 (1979) · Zbl 0433.22008 · doi:10.1016/0001-8708(79)90057-4 |
| [15] | Olshanetsky, M., Perelomov, A.: Classical integrable systems related to Lie algebras. Phys. Rep.71, 313 (1981) · Zbl 0343.70010 · doi:10.1016/0370-1573(81)90023-5 |
| [16] | Flaschka, H.: Introduction to Integrable Systems. Lectures at the AMS Symposium on Theta Functions, Bowdoin 1978, unpublished · Zbl 0396.35076 |
| [17] | Ueno, K., Takasaki, K.: Toda lattice hierarchy. In: Advanced Studies in Pure Math 4, Group Representations and Differential Equations (1984) · Zbl 0577.58020 |
| [18] | David, F.: Loop equations and nonperturbative effects in 2d quantum gravity. Saclay preprint SPhT/90-043 (1990) |
| [19] | Myers, R., Periwal, V.: Exact solution of self-dual unitary matrix models. ITP preprint NSF-ITP-90-73 (1990) · Zbl 1050.82527 |
| [20] | Douglas, M., Seiberg, N., Shenker, S.: Flow and instability in quantum gravity. Rutgers preprint (April 1990) |
| [21] | Moore, G.: Geometry of the string equations. Yale preprint (May 1990) · Zbl 0727.35134 |
| [22] | Itzykson, C., Zuber, J.-B.: The planar approximation. II. J. Math. Phys.21, 411 (1980); Chadha, S., Mahoux, G., Mehta, M. L.: A method of integration over matrix variables. J. Phys.A14, 579 (1981) · Zbl 0997.81549 · doi:10.1063/1.524438 |
| [23] | Leznov, A., Saveliev, M.: Representation of zero curvature for the system of non-linear partial differential equations \(x_{\alpha ,z\bar z} = (e^{k \cdot x} )_\alpha \) and its Integrability. Lett. Math. Phys.3, 489 (1979) · Zbl 0415.35017 · doi:10.1007/BF00401930 |
| [24] | Distler, J.: 2d quantum gravity, topological field theory, and multicritical matrix models. Princeton preprint PUPT-1161 (1989) |
| [25] | Banks, T., Martinec, E.: The renormalization group and string field theory. Nucl. Phys.B294, 733 (1987) · doi:10.1016/0550-3213(87)90605-5 |
| [26] | Eguchi, T., Kawai, H.: Reduction of the dynamical degrees of freedom in the large-N gauge theory. Phys. Rev. Lett.48, 1063 (1982); Bhanot, G., Heller, U., Neuberger, H.: The quenched Eguchi-Kawai model. Phys. Lett.113B, 47 (1982) · doi:10.1103/PhysRevLett.48.1063 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
