Issues

 / 

1994

 / 

January

  

Reviews of topical problems


Integrability and matrix models


Russian Federation State Scientific Center ‘A.I. Alikhanov Institute of Theoretical and Experimental Physics’, ul. Bolshaya Cheremushkinskaya 25, Moscow, 117259, Russian Federation

The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Discrete 1-matrix, 2-matrix, ’conformal’ (multicomponent) and Kontsevich models are considered in some detail, together with the Ward identites (’W-constraints’), determinantal formulas and continuum limits, taking one kind of model into another. Subtle points and directions of the future research are also discussed.

Fulltext pdf (1.5 MB)
Fulltext is also available at DOI: 10.1070/PU1994v037n01ABEH000001
PACS: 02.10.+b, 11.17.+y
DOI: 10.1070/PU1994v037n01ABEH000001
URL: https://ufn.ru/en/articles/1994/1/a/
A1994NA86300001
Citation: Morozov A Yu "Integrability and matrix models" Phys. Usp. 37 1–55 (1994)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Îðèãèíàë: Ìîðîçîâ À Þ «Èíòåãðèðóåìîñòü è ìàòðè÷íûå ìîäåëè» ÓÔÍ 164 3–62 (1994); DOI: 10.3367/UFNr.0164.199401a.0003

References (113) ↓ Cited by (207) Similar articles (20)

  1. Morozov A Usp. Fiz. Nauk 162 (8) 84 (1992); Morozov A Sov. Phys. Usp. 35 671 (1992)
  2. Klebanov I, Polyakov A Mod. Phys. Lett. A 6 3273 (1991)
  3. Witten E Nucl. Phys. B 373 187 (1992)
  4. Marshakov A, Mironov A, Olshanetsky M et al. "c=rG Theories of W G-Gravity: the Set of Observables as a Model of Simply-Laced G" Preprint ITEP-M2/92 hep-th/9203043, to appear in Nucl. Phys. B
  5. Alekseev A, Shatashvili S Nucl. Phys. B 329 719 (1989)
  6. Gerasimov A, Lebedev D et al. Int. J. Mod. Phys. A 6 977 (1991); Morozov A Mod. Phys. Lett. A 6 1525 (1992)
  7. Its A, Izergin A, Korepin V, Slavnov N et al. HEPTH (1992), December
  8. Witten E Comm. Math. Phys. 117 353 (1988); Witten E Comm. Math. Phys. 118 411 (1988)
  9. Witten E Nucl. Phys. B 340 281 (1990); Witten E Surv. Diff. Geom. 1 243 (1991)
  10. Dijkgraaf R, Witten E Nucl. Phys. B 342 486 (1990)
  11. Birmingham D, Blau M, Rakowski M, Thompson G Phys. Rep. 209 129 (1991)
  12. Belavin A, Polyakov A, Zamolodchikov A Nucl. Phys. B 241 333 (1984)
  13. Knizhnik V Usp. Fiz. Nauk 159 (3) 401 (1989); Knizhnik V Sov. Phys. Usp. 32 945 (1989)
  14. Perelomov A i dr. Kompleksnaya Geometriya i Teoriya Strun (M.: VINITI, 1990); English translation, Perelomov A et al. Complex Geometry And String Theory (Berlin: Springer, 1991)
  15. Feigin B, Fuks D Funkts. Analiz Ego Prilozh. 16 (2) 247 (1982); Feigin B, Fuks D Funkts. Analiz Ego Prilozh. 17 (3) 91 (1983); Dotsenko V, Fateev V Nucl. Phys B 240 312 (1984); Dotsenko V, Fateev V Feigin B, Frenkel E Commun. Math. Phys. 128 161 (1990)
  16. Gerasimov A, Marshakov A, Olshanetsky M, Shatashvili S et al. Int. J. Mod. Phys. A 5 2495 (1990)
  17. Losev A "Descendants Constructed from Matter Fields and K Saito Higher Residue Pairing in Topological L-G Theories Coupled to Topological Gravity" Preprint TPI-MINN (1992), May; Losev A, Polyubin I "On Connection between Topological Landau-Ginzburg Gravity and Integrable Systems" preprint ITEP/UU-ITP (1992), December
  18. Migdal A Phys. Rep. 102 199 (1983)
  19. Kazakov V Mod. Phys. Lett. A 4 2125 (1989)
  20. Brezin E, Kazakov V Phys. Lett. B 236 144 (1990); Douglas M, Shenker S Nucl. Phys. B 335 635 (1990); Gross D, Migdal A Phys. Rev. Lett. 64 127 (1990)
  21. Shabat G, Voevodski V Sov. Math. Dokl. 39 38 (1989); Levin A, et al. Phys. Lett. B 243 207 (1990); Smit D-J Commun. Math. Phys. 143 253 (1992)
  22. Kontsevich M Funkts. Analiz Ego Prilozh. 25 (2) 50 (1991)
  23. Witten E Proc. of the NYC Conf., June (1991)
  24. Marshakov A, Mironov A et al. Phys. Lett. B 274 280 (1992)
  25. Kazakov V, Migdal A "Induced QCD at Large N" Preprint PUPT-1322 (1992)
  26. Gerasimov A, Marshakov A, Mironov A, Orlov A, et al. Nucl. Phys. B 357 565 (1991)
  27. Mironov A, et al. Phys. Lett. B 252 47 (1990); Ambjorn J, Jurkiewicz J, Makeenko Yu Phys. Lett. B 251 517 (1990); Itoyama H, Matsuo Y Phys. Lett. B 255 202 (1991)
  28. Makeenko Yu, Marshakov A, Mironov A, et al. Nucl. Phys. B 356 574 (1991)
  29. Fukuma M, Kawai H, Nakayama R Int. J. Mod. Phys. A 6 1385 (1991); Dijkgraaf R, Verlinde E, Verlinde H Nucl. Phys. B 348 565 (1991)
  30. Kharchev S, Marshakov A, Mironov A, Zabrodin A, et al. Phys. Lett. B 275 311 (1992); Kharchev S, Marshakov A, Mironov A, Zabrodin A, et al. Nucl. Phys. B 380 181 (1992)
  31. Dijkgraaf R "Intersection Theory, Integrable Hierarchies and Topological Field Theory" Preprint IASSNS-HEP-91/91; Dijkgraaf R "Intersection Theory, Integrable Hierarchies and Topological Field Theory" hep-th/9201003
  32. Kontsevich M Commun. Math. Phys. 147 (1) 1 (1992)
  33. Adler M, Moerbeke P van Comm. Math. Phys. 147 25 (1992); Adler M, Moerbeke P van Itzykson C, Zuber J-B "Combinatorics of the Modular Group II. The Kontsevich Integrals" Preprint SPhT/92-001, to appear in Int. J. Mod. Phys. A
  34. Crnkovich C, Ginsparg P, Moore G Phys. Lett. B 237 196 (1990); Gross D, Migdal A Phys. Rev. Lett. 64 717 (1990); Brezin E, Douglas M, Kazakov V, Shenker S Phys. Lett. B 237 43 (1990)
  35. Chekhov L, Makeenko Yu Phys. Lett. B 278 271 (1992)
  36. Kharchev S, Marshakov A, Mironov A et al. "Generalized Kontsevich Model versus To da Hierarchy and Discrete Matrix Models" Preprint ITEP-M3/92; Kharchev S, Marshakov A, Mironov A et al. "Generalized Kontsevich Model versus To da Hierarchy and Discrete Matrix Models" hep-th/9203043, to appear in Nucl. Phys. B
  37. Douglas M Phys. Lett. B 238 176 (1990)
  38. Marshakov A, Mironov A et al. Mod. Phys. Lett. A 7 1345 (1992)
  39. Marshakov A, Mironov A et al. Phys. Lett. B 265 99 (1991); Mironov A, Pakuliak S "Double-Scaling Limit in the Matrix Models of a New Type" Preprint FIAN/TD/05-92; Kharchev S, Marshakov A, Mironov A, Pakuliak S et al. "Conformal Matrix Models as an Alternative to Conventional Multi-matrix Models" preprint ITEP-M4/92; Kharchev S, Marshakov A, Mironov A, Pakuliak S et al. "Conformal Matrix Models as an Alternative to Conventional Multi-matrix Models" hep-th/9208044, to appear in Nuc l. Phys. B
  40. Kharchev S, Marshakov A, Mironov A et al. "Landau-Ginzburg Topological Theories in the Framework of GRM and Equivalent Hierarchies" Preprint ITEP-M5/92; Kharchev S, Marshakov A, Mironov A et al. "Landau-Ginzburg Topological Theories in the Framework of GRM and Equivalent Hierarchies" hep-th/9208046, to appear in Mod. Phys. Lett.
  41. Kharchev S, Marshakov A "Topological versus Non-Topological Theories" preprint FIAN/TD-15/92
  42. Date E, Jimbo M, Kashiwara M, Miwa T "Transformation Groups for Soliton Equations" RIMS-394 (1982), February
  43. Witten E "Two-Dimensional Gauge Theories Revisited" Preprint IASSNS-HEP 92/15
  44. Niemi A, Tirkkonen O "On Exact Evaluation of Path Integrals" Preprint UU-ITP 3/93
  45. Gerasimov A, Makeenko Yu, Marshakov A, Mironov A, Orlov A et al. Mod. Phys. Lett. A 6 3079 (1991)
  46. Brezin E, Itzykson C, Parisi G, Zuber J-B Commun. Math. Phys. 59 35 (1978)
  47. Frenkel I, Kac V Inv. Math. 62 23 (1980); Segal G Commun. Math. Phys. 80 301 (1981)
  48. Bershadsky M, Ooguri H Commun. Math. Phys. 126 49 (1989)
  49. Bershadsky M Commun. Math. Phys. 1397 1 (1991)
  50. Bais F A, Tjin T, Driel P van Nucl. Phys. B 357 632 (1991)
  51. Feher L, O’Raifeartaigh L, Ruelle O, Tsutsui I, Wipf A "On the General Structure of H amiltonian Reductions of WZW Theory" Preprint DIAS-STP-91 UdeM-LPN-TH-71/91
  52. Zamolodchikov A Teor. Mat. Fiz. 63 1205 (1985); Lukianov S, Fateev V Int. J. Mod. Phys. A 3 507 (1988)
  53. Alvarez-Gaume L et al. Preprint CERN (1991)
  54. Gross D, Newman M Phys. Lett. B 266 291 (1991)
  55. Mikhailov A Preprint ITEP-UU (1993), March
  56. Chekhov L, Makeenko Yu Phys. Lett. B 278 271 (1992)
  57. Gava E, Narain K Phys. Lett. B 263 213 (1991)
  58. Shifman M, Turbiner A et al. Int. J. Mod. Phys. A 5 2953 (1990)
  59. Migdal A "Exact Solution of Induced Lattice Gauge Theory at Large N" Preprint PUPT-1323 (1992), June
  60. Itzykson C, Zuber J-B J. Math. Phys. 21 411 (1980)
  61. Harish-Chandra Am. J. Math. 79 87 (1957)
  62. Semenov-Tyan-Shanskii M Izv. AN SSSR. Ser. Fiz. 40 562 (1976)
  63. Alekseev A, Faddeev L, Shatashvili S J. Geom. Phys. 3 (1989)
  64. Duistermaat J, Heckman G Invent. Math. 69 259 (1982)
  65. Blau M, Keski-Vakkuri E, Niemi A Phys. Lett. B 246 92 (1990); Keski-Vakkuri E, Niemi A, Semenoff G, Tirkkonen O Phys. Rev. D 44 3899 (1991); Hietamaki A, Niemi A, Palo K et al. Phys. Lett. B 263 417 (1991); Niemi A, Palo K et al. Phys. Lett. B 271 365 (1991); Niemi A, Palo K et al. Nucl. Phys. B 377 295 (1992)
  66. Kogan I, Semenoff G, Weiss N et al. "Area Law and Continuum Limit in "Induced QCD"" Preprint UBCTP92-26/ITEPM6/92; Kogan I, Semenoff G, Weiss N et al. "Area Law and Continuum Limit in "Induced QCD"" hep-th/9208012, to appear in Nucl. Phys. B
  67. Morozov A "Pair Correlator in the Itzykson-Zuber Integral" Preprint ITEP-M10/92 (1992); Morozov A "Pair Correlator in the Itzykson-Zuber Integral" (1992) hep-th/9209074, to appear in Mod. Phys. Lett. A September
  68. Shatashvili S "Correlation Functions in the Itzyks on-Zuber Model" Preprint IASSNS-HEP-92/61 (1992), September
  69. Vinet L et al. "q-Hypergeometric Functions in the Free Field Formalism" Preprint ITEP-M9/92 (1992), August
  70. Miwa T Proc. Jpn. Acad. 58 9 (1982); Saito S Phys. Rev. Lett. 59 1798 (1987); Saito S Phys. Rev. D 36 1819 (1987)
  71. Caselle M, D’Adda A, Panzer S "Exact Solution of D = 1 Kazakov-Migdal Induced Gauge Theory" Preprint DFTT-38/92 (1992), July
  72. Kogan I, Semenoff G, Weiss N et al. "Continuum Limits of "Induced QCD" :Lessons of the Gaussian Model at d=1 and beyond" Preprint ITEP-M7/92, UBCTP 92-27 (1992), to appear in Int. J. Mod. Phys. A August
  73. Olshanetsky M, Perelomov A Phys. Rep. 71 (5) (1981)
  74. Freund P, Zabrodin A Phys. Lett. B 284 283 (1992); Freund P, Zabrodin A Commun. Math. Phys. 147 277 (1992); Freund P, Zabrodin A Zabrodin A Mod. Phys. Lett. A 7 441 (1992)
  75. Freund P, Zabrodin A "Excitation Scattering in Integrable Models and Hall-Littlewood-Kerov Polynomials" hep-th/9208063
  76. Turbiner A Funkts. Analiz Ego Prilozh. 22 33 (1988); Turbiner A Commun. Math. Phys. 118 467 (1988); Shifman M, Turbiner A Commun. Math. Phys. 120 347 (1989)
  77. Perelomov A, Rosly A, Shifman M, Turbiner A et al. Int. J. Mod. Phys. A 5 803 (1990)
  78. Halpern M, Kiritsis E Mod. Phys. Lett. A 4 1973 (1989); Halpern M, Kiritsis E, Obers N, Porrati M, Yamron J Int. J. Mod. Phys. A 5 2275 (1990)
  79. Niemi A, Pasanen P Phys. Lett. B 253 349 (1991)
  80. Polychronakos A Nucl. Phys. B 324 597 (1989); Leinaas J, Myrheim J Phys. Rev. B 37 9286 (1988); Brink L, Hansson T, Konstein S, Vasiliev M "The Calogero Model-Anyonic Representation, Fermionic Extension and Super symmetry" Preprint USITP-92-14/Goteborg ITP-92-53 (1993), January
  81. Gorsky A, Nekrasov N "Hamiltonian Systems of Calogero Type and Two-Dimensional Yang-Mills Theory" Preprint ITEP-20/93, UUITP-6/93 (1993), March
  82. Sato M RIMS Kokyuroku 439 30 (1981); Sato M, Sato Ya Lect. Notes Num. Appl. Anal. 5 259 (1982); Segal G, Wilson G Publ. I.H.E.S. 61 1 (1985)
  83. Kac V Infinite Dimensional Lie Algebras (Cambridge: Cambridge University Press, 1985)
  84. Orlov A, Shulman E Lett. Math. Phys. 12 171 (1986); Grinevich P, Orlov A "Problems of Mo dern Quantum Field Theory" Proceedings of Kiev Conference (Eds A Belavin et al.) (Berlin: Springer, 1989)
  85. de Voos K Nucl. Phys. B 375 478 (1992)
  86. Hirota R "Direct Methods in Soliton Theory" Topics In Current Physics Vol. 17 (Eds R Bullough, P Caudrey) (New York: Springer, 1980) p. 157
  87. Friedan D, Shenker S Phys. Lett. B 175 287 (1986); Ishibashi N, Matsuo Y, Ooguri H Mod. Phys. Lett. A 2 119 (1987); Alvarez-Gaume L, Gomez C, Reina C Phys. Lett. B 190 55 (1987); Witten E Commun. Math. Phys. 113 529 (1988); Morozov A Phys. Lett. B 196 325 (1987)
  88. Jimbo M, Miwa T, Sato M Holonomic Quantum Fields I-V Vol. 14 (Kyoto: RIMS, Kyoto Univ., 1978) p. 223; Jimbo M, Miwa T, Sato M Holonomic Quantum Fields I-V 15 (Kyoto: RIMS, Kyoto Univ.) 201, 577, 871, 1531 (1979)
  89. Kharchev S, Marshakov A, Mironov A, Orlov A, Zabrodin A Nucl. Phys. B 366 569 (1991)
  90. Ueno K, Takasaki K Adv. Stud. Pure Math. 4 1 (1984)
  91. Bowick M, Shewitz D et al. Nucl. Phys. B 354 496 (1991)
  92. Hirota R, Ohta Y, Satsuma J J. Phys. Soc. Jpn. 57 1901 (1988)
  93. Kac V, van de Leur J "The n-Component KP Hierarchy and Representation Theory" Preprint (1993), January
  94. Moore G Commun. Math. Phys. 133 261 (1990)
  95. Witten E "The N Matrix Model and Gauged WZW Models" Preprint IAS-HEP-91/26, to appear in to appear in Nucl. Phys. B
  96. Krichever I Commun. Math. Phys. 143 415 (1992); Dubrovin B Nucl. Phys. B 379 627 (1992)
  97. Takasaki K, Takebe K "S Diff (2) KP Hierarchy" Preprint RIMS-814 (1991), October
  98. Makeenko Yu, Semenoff G Int. J. Mod. Phys. A 6 3455 (1991); Gross D, Newman M Phys. Lett. B 266 291 (1991)
  99. Di Francesco P, Itzykson C, Zuber J-B Commun. Math. Phys. 151 193 (1993)
  100. Takebe T "From General Zakharov-Shabat Equations to the KP and the Toda-Lattice Hierarchies" Preprint RIMS-779 (1991)
  101. Date E, Jimbo M, Kashiwara M, Miwa T Transformation Groups for Soliton Equations RIMS-394 (1982), February
  102. Kharchev S et al. Preprint ITEP-M12/92
  103. Novikov S, Manakov S, Pitaevski L, Zakharov V Theory Of Solitons (New York: Plenum, 1984)
  104. Fairlie D, Govaerts J et al. Nucl. Phys. B 373 214 (1992); Fairlie D, Govaerts J Phys. Lett. B 281 49 (1992); Fairlie D, Govaerts J J. Math. Phys. 33 3542 (1992)
  105. Makeenko Yu, Migdal A Phys. Lett. B 88 135 (1979)
  106. Penner R Commun. Math. Phys. 113 299 (1987); Penner R J. Differ. Geom. 27 35 (1988)
  107. Martinec E Phys. Lett. B 217 431 (1989); Martinec E "Criticalities, catastrophe and compactigications" V Knizhnik Memorial Volume (1980); Vafa C, Warner N Phys. Lett. B 218 51 (1989); Greene B, Vafa C, Warner N Nucl. Phys. B 324 371 (1989); Lerche W, Vafa C, Warner N Nucl. Phys. B 324 427 (1989); Vafa C Mod. Phys. Lett. A 6 337 (1990); Dijkgraaf R, Verlinde E, Verlinde H Nucl. Phys. B 352 59 (1991); Li K Nucl. Phys. B 354 711 (1991)
  108. Elitzur S, Forge A, Rabinovici E Nucl. Phys. B 388 131 (1992)
  109. Cecotti S, Vafa C Nucl. Phys. B 367 359 (1991)
  110. Kac V, Schwarz A Phys. Lett. B 257 329 (1991)
  111. Semenoff G, Weiss N "Symmetry and Observables in Induced QCD" Preprint UBC (1993), March; to appear in, Semenoff G, Weiss N Proc. of Rakhov School on Mathematical Physics, String Theory and Quantum Gravity, Rakhov, Ukraine, October 1992
  112. Klebanov I Proc. of 1991 Trieste Spring School on Strings and Quantum Gravity (Ed. J Harvey et al.) (Singapore: World Scientific, 1991); Verlinde E, Verlinde H "A Unitary S-Matrix for 2d Black Hole Formation and Evaporation" Preprint PUPT-1380, IASSNS-HEP-93/8 (1993), February
  113. Dijkgraaf R, Moore G, Plesser R "The Partition Function of 2D String Theory" Preprint IASSNS-HEP-92/48, YCTP-P22-92; Dijkgraaf R, Moore G, Plesser R "The Partition Function of 2D String Theory" hep-th/9208031, to appear in Nucl. Phys. B

© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions