String-orthogonal polynomials, string equations, and 2-toda symmetries

Communications on Pure and Applied Mathematics

Volumn 50, Issue 3, 1997, Pages 241-290

  Adler, Mark ^a   Van Moerbeke, Pierre ^a,b  

^a BRANDEIS UNIVERSITY   (United States)

^b UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0005921172     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1097-0312(199703)50:3<241::AID-CPA3>3.0.CO;2-B     Document Type: Article

References (38)

1. Adler, M., Morozov, A., Shiota, T., and van Moerbeke, P., A matrix integral solution to [P, Q] = P and matrix Laplace transforms, Comm. Math. Phys. 180, 1996, pp. 233-263.
2. p-gravity, Comm. Math. Phys. 147, 1992, pp. 25-56.
3. Adler, M., and van Moerbeke, P., Matrix integrals, Toda symmetries, Virasoro constraints and orthogonal polynomials, Duke Math. J. 80, 1995, pp. 863-911.
4. Adler, M., and van Moerbeke, P., Lie algebra splittings, 2-Toda and string-orthogonal polynomials, Internat. Math. Res. Notices, 1997, to appear.
5. ∞-algebra to its central extension: A τ-function approach, Phys. Lett. A 194, 1994, pp. 33-43.
6. Adler, M., Shiota, T., and van Moerbeke, P., A Lax representation for the vertex operator and the central extension, Comm. Math. Phys. 171, 1995, pp. 547-588.
7. Adler, M., Shiota, T., and van Moerbeke, P., Random matrices, vertex operators and the Virasoro algebra, Phys. Lett. A 208, 1995, pp. 67-78.
8. Adler, M., Shiota, T., and van Moerbeke, P., The spectrum of coupled random matrices, 1997, to appear.
9. Adler, M., Shiota, T., and van Moerbeke, P., Grassmannian pairings, solitons and string equations, 1997, to appear.
10. Ahn, Ch., and Shigemoto, K., Multi-matrix model and 2D Toda multi-component hierarchy, Phys. Lett. B 263, 1991, pp. 44-50.
11. Ahn, Ch., and Shigemoto, K., One-point functions of loops and constraint equations of the multi-matrix models at finite N, Phys. Lett. B 285, 1992, pp. 42-48.
12. Alvarez-Gaumé, L., Gomez, C., and Lacki, J., Integrability in random matrix models, Phys. Lett B 253, 1991, pp. 56-62.
13. Brézin, É., and Kazakov, V. A., Exactly solvable field theories of closed strings, Phys. Lett. B 236, 1990, pp. 144-150.
14. Chadha, S., Mahoux, G., and Mehta, M. L., A method of integration over matrix variables. II, J. Phys. A 14, 1981, pp. 579-586.
15. Date, E., Kashiwara, M., Jimbo, M., and Miwa, T., Transformation groups for soliton equations, pp. 39-119 in: Nonlinear Integrable Systems - Classical Theory and Quantum Theory (Kyoto 1981), World Scientific, Singapore, 1983.
16. Dickey, L. A., Soliton Equations and Hamiltonian Systems. Advanced Series in Mathematical Physics 12, World Scientific, River Edge, N.J., 1991.
17. Dickey, L. A., Additional symmetries of KP, Grassmannian, and the string equation, Modern Phys. Lett. A 8, 1993, pp. 1259-1272.
18. Dickey, L. A., On additional symmetries of the KP hierarchy and Sato's Bäcklund transformation, Comm. Math. Phys. 167, 1995, pp. 227-233.
19. Douglas, M. R., and Shenker, S. H., Strings in less than one dimension, Nuclear Phys. B 335, 1990, pp. 635-654.
20. Fuchssteiner, B., Mastersymmetries, higher order time-dependent symmetries and conserved densities of nonlinear evolution equations, Progr. Theoret. Phys. 70, 1983, pp. 1508-1522.
21. Gerasimov, A., Marshakov, A., Mironov, A., Morozov, A., and Orlov, A., Matrix models of two-dimensional gravity and Toda theory, Nuclear Phys. B 357, 1991, pp. 565-618.
22. Gross, D. J., and Migdal, A. A., Non-perturbative two-dimensional quantum gravity, Phys. Rev. Lett. 64, 1990, pp. 127-130.
23. Kac, V. G., Infinite-Dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
24. Kac, V., and Raina, A. K., Bombay Lectures on Highest Weight Representations of Infinite-Dimensional Lie Algebras. Advanced Series in Mathematical Physics No. 2, World Scientific, Teaneck, N.J., 1987.
25. Mehta, M. L., A method of integration over matrix variables, Comm. Math. Phys. 79, 1981, pp. 327-340.
26. Martinec, E. J., On the origin of integrability in matrix models, Comm. Math. Phys. 138, 1991, pp. 437-449.
27. Morozov, A., Integrability and matrix models, Uspekhi Fiz. Nauk 164(1), 1994, pp. 3-62.
28. Moore, G., Geometry of the string equations, Comm. Math. Phys. 133, 1990, pp. 261-304.
29. Mulase, M., Algebraic theory of the KP equations, pp. 151-217 in: Perspectives in Mathematical Physics, Conference Proceedings and Lecture Notes in Mathematical Physics III, International Press, Cambridge, MA, 1994.
30. Mulase, M., Matrix integrals and integrable systems, pp. 111-127 in: Topology, Geometry and Field Theory, World Scientific, River Edge, N.J., 1994.
31. Orlov, A. Yu., and Schulman, E. I., Additional symmetries for integrable equations and conformal algebra representation, Lett. Math. Phys. 12, 1986, pp. 171-179.
32. Szegö, G., Orthogonal Polynomials, 4th ed., American Mathematical Society, Colloquium Publications, Vol. 23, American Mathematical Society, Providence, R.I., 1975.
33. Ueno, K., Takasaki, K., Toda lattice hierarchy, pp. 1-95 in: Group Representations and Systems of Differential Equations (Tokyo, 1982), K. Okamoto, ed., Advanced Studies in Pure Mathematics No. 4, North-Holland, Amsterdam-New York, 1984.
34. van de Leur, J. W., The Adler-Shiota-van Moerbeke formula for the BKP hierarchy, J. Math. Phys. 36, 1995, pp. 4940-4951.
35. 1+∞ constraints, J. Geom. Phys. 17, 1995, pp. 95-124.
36. van Moerbeke, P., Integrable foundations of string theory, pp. 163-267 in: Integrable Systems. The Verdier Memorial Conference Proceedings of the International Conference held in Luminy, July 1-5, 1991, O. Babelon, P. Cartier, and Y. Kosmann-Schwarzbach, eds., Progress in Mathematics No. 115, Birkhäuser Boston, Boston, 1993.
37. van Moerbeke, P., and Mumford, D., The spectrum of difference operators and algebraic curves, Acta Math. 143, 1979, pp. 93-154.
38. Witten, E., Two-dimensional gravity and intersection theory on moduli space, pp. 243-310 in: Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh University, Bethlehem, Pa., 1991.