Exact correlators of two-matrix models

Journal of Geometry and Physics

Volumn 20, Issue 2-3, 1996, Pages 160-194

  Bonora, L ^a   Constantinidis, C P ^a   Xiong, C S ^b  

^a SEZIONE DI TRIESTE   (Italy)

^b UNIVERSITY OF TOKYO   (Japan)

Author keywords

Correlation function; Matrix model; p KdV hierarchy; Toda lattice hierarchy; Topological field theory

Indexed keywords

EID: 0030268926     PISSN: 03930440     EISSN: None     Source Type: Journal    
DOI: 10.1016/0393-0440(95)00052-6     Document Type: Article

References (42)

1. [1] M. Douglas, Strings in less than one dimension and the generalized KdV hierarchies, Phys. Lett. B 238 (1990) 176; The two matrix model in: Random Surfaces and Quantum Gravity, ed. O. Alvarez et al. (Cargèse, 1990) 77-83.
2. [1] M. Douglas, Strings in less than one dimension and the generalized KdV hierarchies, Phys. Lett. B 238 (1990) 176; The two matrix model in: Random Surfaces and Quantum Gravity, ed. O. Alvarez et al. (Cargèse, 1990) 77-83.
3. [2] E. Brezin, M. Douglas, V. Kazakov and S. Shenker, The Ising model coupled to 2D gravity. A nonperturbative analysis, Phys. Lett. B 237 (1990) 43; D.J. Gross and A.A. Migdal, Nonperturbative solution of the Ising model on a random surface, Phys. Rev. Lett. 64 (1990) 717.
4. [2] E. Brezin, M. Douglas, V. Kazakov and S. Shenker, The Ising model coupled to 2D gravity. A nonperturbative analysis, Phys. Lett. B 237 (1990) 43; D.J. Gross and A.A. Migdal, Nonperturbative solution of the Ising model on a random surface, Phys. Rev. Lett. 64 (1990) 717.
5. [3] C. Crnkovic, P. Ginsparg and G. Moore, The Ising model, the Yang-Lee edge singularity and 2-D quantum gravity, Phys. Lett. B 237 (1990) 196.
6. [4] J. Goeree, W-constraints in 2D quantum gravity, Nuclear Phys. B 358 (1991) 737.
7. 3, algebra, Phys. Lett. B 263 (1991) 213; E. Martinec, On the origin of integrability in matrix models, Comm. Math. Phys. 138 (1991) 437.
8. 3, algebra, Phys. Lett. B 263 (1991) 213; E. Martinec, On the origin of integrability in matrix models, Comm. Math. Phys. 138 (1991) 437.
9. [6] T. Tada and T. Yamaguchi, P and Q operator analysis for the two-matrix model, Phys. Lett. B 250 (1990) 38; T. Tada, (p,q) critical point from two-matrix model, Phys. Lett. B 259 (1991) 442; S. Odake, Two-matrix model and minimal unitary model, Phys. Lett. B 269 (1991) 300.
10. [6] T. Tada and T. Yamaguchi, P and Q operator analysis for the two-matrix model, Phys. Lett. B 250 (1990) 38; T. Tada, (p,q) critical point from two-matrix model, Phys. Lett. B 259 (1991) 442; S. Odake, Two-matrix model and minimal unitary model, Phys. Lett. B 269 (1991) 300.
11. [6] T. Tada and T. Yamaguchi, P and Q operator analysis for the two-matrix model, Phys. Lett. B 250 (1990) 38; T. Tada, (p,q) critical point from two-matrix model, Phys. Lett. B 259 (1991) 442; S. Odake, Two-matrix model and minimal unitary model, Phys. Lett. B 269 (1991) 300.
12. [7] J. De Boer, Multi-matrix models and the KP hierarchy, Nuclear Phys. B 336 (1991) 602.
13. [8] S. Ryang, Genus one correlation functions and recursion relations in the two-matrix model, Phys. Rev. D 46 (1992) 1873.
14. [9] C. Ahn and K. Shigemoto, Multi-matrix model and 2D Toda multi-component hierarchy, Phys. Lett. B 263 (1991) 44.
15. [10] S. Karchev, A. Marshakov, A. Mironov, A. Morozov and A. Zabrodin, Matrix models among integrable hierarchies: forced hierarchies and operator formalism, Nuclear Phys. B 366 (1991) 569; A. Marshakov, A. Mironov and A. Morozov, From Virasoro constraints in Kontsevitch's model to W constraints in two-matrix models, Modern Phys. Lett. A 7 (1992) 1345.
16. [10] S. Karchev, A. Marshakov, A. Mironov, A. Morozov and A. Zabrodin, Matrix models among integrable hierarchies: forced hierarchies and operator formalism, Nuclear Phys. B 366 (1991) 569; A. Marshakov, A. Mironov and A. Morozov, From Virasoro constraints in Kontsevitch's model to W constraints in two-matrix models, Modern Phys. Lett. A 7 (1992) 1345.
17. [11] I.K. Kostov, Gauge invariant matrix models and the Â-D̂-Ê closed strings, Phys. Lett. B 297 (1992) 74.
18. [12] S. Panda and S. Roy, Remarks on the additional symmetries and W-constraints in the generalized KdV hierarchy, Phys. Lett. B 296 (1992) 23.
19. [13] J.-M. Daul, V.A. Kazakov and I.K. Kostov, Rational theories of 2d gravity from the two-matrix model, Nuclear Phys. B 409 (1993) 311.
20. [14] M. Staudacher, Combinatorial solutions of the two-matrix model, Phys. Lett. B 305 (1993) 332.
21. [15] L. Bonora and C.S. Xiong, Multi-matrix models without continuum limit, Nuclear Phys. B 405 (1993) 191.
22. [16] L. Bonora and C.S. Xiong, Extended Toda lattice hierarchy, extended two-matrix model and c = 1 string theory, Nuclear Phys. B 434 (1995) 408.
23. [17] D. Bessis, C. Itzykson, G. Parisi and J.-B. Zuber, Planar diagrams, Comm. Math. Phys. 59 (1978) 35; D. Bessis, C. Itzykson and J.-B. Zuber, Quantum field theory techniques in graphical enumeration, Adv. Appl. Math. 1 (1980) 109.
24. [17] D. Bessis, C. Itzykson, G. Parisi and J.-B. Zuber, Planar diagrams, Comm. Math. Phys. 59 (1978) 35; D. Bessis, C. Itzykson and J.-B. Zuber, Quantum field theory techniques in graphical enumeration, Adv. Appl. Math. 1 (1980) 109.
25. [18] C. Itzykson and B. Zuber, The planar approximation. II, J. Math. Phys. 21 (1980) 411.
26. [19] M.L. Mehta, A method of integration over matrix variables, Comm. Math. Phys. 79 (1981) 327; S. Chedha, G. Mahoux and M. Mehta, A method of integration over matrix variables: II, J. Phys. A: Math. Gen. 14 (1981) 579.
27. [19] M.L. Mehta, A method of integration over matrix variables, Comm. Math. Phys. 79 (1981) 327; S. Chedha, G. Mahoux and M. Mehta, A method of integration over matrix variables: II, J. Phys. A: Math. Gen. 14 (1981) 579.
28. [20] K. Ueno and K. Takasaki, Toda lattice hierarchy, Adv. Stud. Pure Math. 4 (1984) 1.
29. [21] H. Aratyn, E. Nissimov, S. Pacheva and A.H. Zimerman, Two-matrix string model as constrained (2+1)-dimensional integrable system, Phys. Lett. B 341 (1994) 19; H. Aratyn, E. Nissimov and S. Pacheva, Darboux-Bäklund solutions of SL (p,q) KP-KdV hierarchies, constrained generalized Toda lattices and two-matrix string model, BGU-94/25, UICHEP-TH/94-12, hepth/9501018.
30. [21] H. Aratyn, E. Nissimov, S. Pacheva and A.H. Zimerman, Two-matrix string model as constrained (2+1)-dimensional integrable system, Phys. Lett. B 341 (1994) 19; H. Aratyn, E. Nissimov and S. Pacheva, Darboux-Bäklund solutions of SL (p,q) KP-KdV hierarchies, constrained generalized Toda lattices and two-matrix string model, BGU-94/25, UICHEP-TH/94-12, hepth/9501018.
31. [22] L. Bonora and C.S. Xiong, Matrix models, topological field theories and c ≤ 1 string theory, Progr. Theoret. Phys. Suppl. 118 (1995) 201; Proc. Kyoto Conf. on Quantum Field Theory, Integrable Models and Beyond (Kyoto 1994); L. Bonora, Two-matrix models, W algebras and 2D gravity, SISSA-170/94/EP, Varenna Lecture Notes, (1994).
32. [22] L. Bonora and C.S. Xiong, Matrix models, topological field theories and c ≤ 1 string theory, Progr. Theoret. Phys. Suppl. 118 (1995) 201; Proc. Kyoto Conf. on Quantum Field Theory, Integrable Models and Beyond (Kyoto 1994); L. Bonora, Two-matrix models, W algebras and 2D gravity, SISSA-170/94/EP, Varenna Lecture Notes, (1994).
33. [22] L. Bonora and C.S. Xiong, Matrix models, topological field theories and c ≤ 1 string theory, Progr. Theoret. Phys. Suppl. 118 (1995) 201; Proc. Kyoto Conf. on Quantum Field Theory, Integrable Models and Beyond (Kyoto 1994); L. Bonora, Two-matrix models, W algebras and 2D gravity, SISSA-170/94/EP, Varenna Lecture Notes, (1994).
34. [23] L. Bonora and C.S. Xiong, Topological field theory interpretations and LG representation of c = 1 string theory, Nucl. Phys. B. 453 (1995) 129.
35. [24] L. Bonora and C.S. Xiong, Multi-field representation of the KP hierarchy and multi-matrix models, Phys. Lett. B 317 (1993) 329.
36. [25] L.A. Dickey, Soliton Equations and Hamiltonian Systems (World Scientific, Singapore, 1991).
37. [26] L. Bonora and C.S. Xiong, The (N, M)th KdV hierarchy and the associated algebras, J. Math. Phys. 35 (1994) 5781.
38. [27] L. Bonora, Q.P. Liu and C.S. Xiong, The integrable hierarchy constructed from a pair of KdV-type hierarchies and its associated W algebra, hep-th/9408035, Comm. Math. Phys., to be published.
39. [28] L. Bonora and C.S. Xiong, Matrix models without scaling limit, Internat. J. Modern Phys. A 8 (1993) 2973.
40. [29] M. Douglas and S. Shenker, Strings in less than one dimension, Nuclear Phys. B 335 (1990) 635.
41. [30] S. Aoyama and Y. Kodama, Topological conformal field theory with a rational W potential and the dispersionless KP hierarchy, Modern Phys. Lett. A 9 (1994) 2481; Topological Landau-Ginzburg theory with a rational W potential and the dispersionless KP hierarchy, Ohio State University Math. Research Institute, preprint 5/22/95, hep-th/9505122.
42. [30] S. Aoyama and Y. Kodama, Topological conformal field theory with a rational W potential and the dispersionless KP hierarchy, Modern Phys. Lett. A 9 (1994) 2481; Topological Landau-Ginzburg theory with a rational W potential and the dispersionless KP hierarchy, Ohio State University Math. Research Institute, preprint 5/22/95, hep-th/9505122.