Matrix Integrals, Toda Symmetries, Virasoro Constraints and Orthogonal Polynomials

Adler, M.; van Moerbeke, P.

Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Tome 47 (1995), Exposé no. 9, 48 p.

[AGL] Alvarez-Gaumé, L., Gomez, C., Lacki, J. : Integrability in random matrix models, Phys. Letters B, vol 253, (1,2), 56-62. | MR

[A-S-V] Adler, M., Shiota, T., Van Moerbeke, P. : From the $w_{\infty }$-algebra to its central extension: a $\tau$-function approach, Phys. Letters A, 1994 | MR | Zbl

Adler, M., Shiota, T., Van Moerbeke, P. and A Lax pair representation for the Vertex operator and the central extension, Comm. Math. Phys. 1995. | MR | Zbl

[A-vM2] Adler, M., Van Moerbeke, P. : A matrix integral solution to two-dimensional Wp-Gravity, Comm. Math. Phys. 147, 25-56 (1992). | MR | Zbl

[A-S] Al-Salam, W. A., Chihara, T. S.: Another characterization of the classical orthogonal polynomials, SIAM J. Math. Anal. 3, pp 65-70 (1972). | MR | Zbl

[BIZ] Bessis, D., Itzykson, Cl., Zuber, J.-B. : Quantum field theory techniques in graphical enumeration, Adv. Appl. Math. 1, 109-157 (1980). | MR | Zbl

[B-K] Brezin, E., Kazakov, V., Phys. Lett., 236 B 144 (1990). | MR

[C] Chihara, T.: An introduction to orthogonal polynomials, Gordon and Breach, NY, 1978. | MR | Zbl

[DJKM] Date, E., Jimbo, M., Kashiwara, M., Miwa, T. : Transformation groups for soliton equations, Proc. RIMS Symp. Nonlinear integrable systems, Classical and quantum theory (Kyoto 1981), pp. 39-119. Singapore : World Scientific 1983. | MR | Zbl

[D1] Dickey, L. : Soliton equations and integrable systems, World Scientific (1991). | MR

[D2] Dickey, L. : Additional symmetries of KP, Grassmannian, and the string equation, preprint 1992. | MR | Zbl

[D] Dijkgraaf, R. : Intersection Theory, Integrable Hierarchies and Topological Field Theory, Lecture notes at Cargese Summer school (1991). | MR | Zbl

[Fa] Fastré, J. : Boson-correspondence for $W$-algebras, Bäcklund-Darboux transformations and the equation $\left|L,P\right|=L^n$, University of Louvain, doctoral dissertation (1993)

[GMMMO] Gerasimov, A., Marshakov, A., Mironov, A., Morozov, A., Orlov, A. : Matrix models of Two-dimensional gravity and Toda theory, Nuclear Physics B357, 565-618 (1991). | MR

[G] Ginsparg, P. : Lectures given at Trieste Summer school (july 1991).

[GOS] Grinevich, P. G., Orlov, A. Y., Schulman E. I. : On the symmetries of integrable systems, Important developments in soliton theory, Eds. A. S. Fokas, V. E. Zakharov, Springer-Verlag series in non-linear dynamics, p. 283-301 (1993). | MR | Zbl

[G-M] Gross, D., Migdal, A. :Phys. Rev. Lett. 64, p. 127 (1990). | MR | Zbl

[G-N] Gross, D. J., Newman, M. J. : Unitary and hermitian matrices in an external field (II). The Kontsevich model and continuum Virasoro constraints, Nucl. Phys. B "80 (1992) 168-180. | MR

[H-H] Haine, L., Horozov, E. : Toda Orbits of Laguerre Polynomials and Representations of the Virasoro Algebra, Bulletin des Sciences Math. (1993). | MR | Zbl

[K-R] Kac, V. G., Raina, A. K. : Bombay lectures on highest weight representations of infinite dimensional Lie Algebras, Adv. Series Math. Phys. vol. 2, 1987. | MR | Zbl

[K-S] Karlin, S., Szegö, G.: On certain determinants whose elements are orthogonal polynomials, J. d'An. Math. 8, pp 1-157 (1961). | MR | Zbl

[Kr] Krichever, I. : On Heisenberg Relations for the Ordinary Linear Differential Operators (1990).

[Mal] Martinec, E. J. : On the Origin of Integrability in Matrix Models, Comm. Math. Phys. 138, 437-449 (1991) | MR | Zbl

[Ma2] Martinec, E. J. : An introduction to 2d Gravity and Solvable String Models (lecture notes at Trieste spring school 1991)

[M-M] Mironov, A., Morozov, A. : On the origin of Virasoro constraints in matrix models : Lagrangian approach, Phys. Letters B vol 252, number 1, 47-52 | MR

[O-Sc] Orlov, A. Y., Schulman, E. I. : Additional Symmetries for Integrable and Conformal Algebra Representation, Letters in Math. Phys., 12 171-179 (1986) | MR | Zbl

[U-T] Ueno, K., Takasaki, K. : Toda Lattice Hierarchy, Adv. Studies in Pure Math. 4, (1984) 1-95. | MR | Zbl

[vM2] Van Moerbeke, P.: Integrable foundations of string theory, in Lecures on Integrable systems, Proceedings of the CIMPA-school, 1991, Ed.: O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach, World scientific, pp 163-267 (1994). | MR | Zbl

[VDL1] Van De Leur, J. W. : The Adler-Shiota-van Moerbeke formula (preprint 1994)

[VDL2] Van De Leur, J. W. : (preprint 1994)

[W1] Witten, Ed. : Two-dimensional gravity and intersection theory of moduli space, Harvard University lecture; May 1990, Journal of diff. geometry 1991. | MR | Zbl