Matrix Models: Geometry of Moduli Spaces and Exact Solutions

L. O. Chekhov

L. O. Chekhov

Teoretičeskaâ i matematičeskaâ fizika, Tome 127 (2001) no. 2, pp. 179-252

Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Résumé

We study the connection between characteristics of moduli spaces of Riemann surfaces with marked points and matrix models. The Kontsevich matrix model describes intersection indices on continuous moduli spaces, and the Kontsevich–Penner matrix model describes intersection indices on discretized moduli spaces. Analyzing the constraint algebras satisfied by various generalized Kontsevich matrix models, we derive time transformations that establish exact relations between different models appearing in mathematical physics. We solve the Hermitian one-matrix model using the moment technique in the genus expansion and construct a recursive procedure for solving this model in the double scaling limit.

Export
Comment citer

@article{TMF_2001_127_2_a0,
     author = {L. O. Chekhov},
     title = {Matrix {Models:} {Geometry} of {Moduli} {Spaces} and {Exact} {Solutions}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {179--252},
     year = {2001},
     volume = {127},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2001_127_2_a0/}
}

TY  - JOUR
AU  - L. O. Chekhov
TI  - Matrix Models: Geometry of Moduli Spaces and Exact Solutions
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2001
SP  - 179
EP  - 252
VL  - 127
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2001_127_2_a0/
LA  - ru
ID  - TMF_2001_127_2_a0
ER  -

%0 Journal Article
%A L. O. Chekhov
%T Matrix Models: Geometry of Moduli Spaces and Exact Solutions
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2001
%P 179-252
%V 127
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2001_127_2_a0/
%G ru
%F TMF_2001_127_2_a0

L. O. Chekhov. Matrix Models: Geometry of Moduli Spaces and Exact Solutions. Teoretičeskaâ i matematičeskaâ fizika, Tome 127 (2001) no. 2, pp. 179-252. http://geodesic.mathdoc.fr/item/TMF_2001_127_2_a0/

Bibliographie
Cité par

[1] A. M. Polyakov, Phys. Lett. B, 103 (1981), 207 ; 211 | DOI | MR | MR

[2] J. Polchinsky, Nucl. Phys. B, 324 (1989), 123 | DOI | MR

[3] V. Knizhnik, A. Polyakov, A. Zamolodchikov, Mod. Phys. Lett. A, 3 (1988), 819 | DOI | MR

[4] F. David, Mod. Phys. Lett. A, 3 (1988), 1651 | DOI | MR

[5] J. Distler, H. Kawai, Nucl. Phys. B, 312 (1989), 509 | DOI | MR

[6] J. Ambjørn, B. Durhuus, J. Fröhlich, P. Orland, Nucl. Phys. B, 270 (1986), 457 | DOI | MR

[7] E. Witten, Nucl. Phys. B, 340 (1990), 281 | DOI | MR

[8] E. Witten, “On the Kontsevich model and other models of two-dimensional gravity”, Proc. of the XX Int. Conf. on Diff. Geom. Methods in Theor. Phys., V. 1 (New York, 1991), eds. S. Catto and A. Rocha, World Scientific, River Edge, N.J., 1992, 176 | MR | Zbl

[9] M. L. Kontsevich, Funkts. analiz i ego prilozh., 25 (1991), 50 ; M. L. Kontsevich, Commun. Math. Phys., 147 (1992), 1 | MR | Zbl | DOI | MR | Zbl

[10] G.'t Hooft, Nucl. Phys. B, 72 (1974), 461 | DOI

[11] P. Di Francesco, P. Ginsparg, J. Zinn-Justin, Phys. Rep., 254 (1995), 1 ; E-print hep-th/9306153 | DOI | MR

[12] A. Yu. Morozov, UFN, 164 (1994), 3 | DOI

[13] M. L. Mehta, Random Matrices, Academic Press, New York, 1991 | MR | Zbl

[14] J. Ambjørn, B. Durhuus, T. Jonsson, Quantum Geometry. A Statistical Field Theory Approach, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[15] E. Brézin, C. Itzykson, G. Parisi, J.-B. Zuber, Commun. Math. Phys., 59 (1978), 35 | DOI | MR | Zbl

[16] E. Brézin, V. Kazakov, Phys. Lett. B, 236 (1990), 144 | DOI | MR

[17] M. Douglas, S. Shenker, Nucl. Phys. B, 335 (1990), 635 | DOI | MR

[18] D. J. Gross, A. Migdal, Phys. Rev. Lett., 64 (1990), 127 | DOI | MR | Zbl

[19] A. D. Mironov, $\tau$-Funktsii i matrichnye modeli, Diss. ...dokt. f.-m. n., FIAN, M., 1997

[20] S. Kharchev, Am. Math. Soc. Transl. (2), 191 (1999), 119 | MR | Zbl

[21] E. Witten, Surv. Diff. Geom., 1 (1991), 243 | DOI | MR

[22] V. Kazakov, I. Kostov, N. Nekrasov, Nucl. Phys. B, 557 (1999), 413 ; E-print hep-th/9810035 | DOI | MR | Zbl

[23] T. Banks, W. Fischler, S. H. Shenker, L. Susskind, Phys. Rev. D, 55 (1997), 5112 ; E-print hep-th/9610043 | DOI | MR

[24] N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya, Nucl. Phys. B, 498 (1997), 467 ; E-print hep-th/9612115 | DOI | MR | Zbl

[25] A. B. Zamolodchikov, Pisma v ZhETF, 46 (1987), 129 | MR

[26] A. Yu. Morozov, UFN, 162:8 (1992), 83 | DOI

[27] V. Knizhnik, UFN, 159:3 (1989), 401 | DOI | MR

[28] S. Saito, Phys. Rev. Lett., 59 (1987), 1798 | DOI | MR

[29] S. Saito, Phys. Rev. D, 36 (1987), 1819 | DOI | MR

[30] A. Morozov, Phys. Lett. B, 196 (1987), 325 | DOI | MR

[31] M. Fukuma, H. Kawai, R. Nakayama, Int. J. Mod. Phys. A, 6 (1991), 1385 | DOI | MR

[32] A. Mironov, A. Morozov, Phys. Lett. B, 252 (1990), 47 | DOI | MR

[33] J. Ambjørn, J. Jurkiewicz, Yu. Makeenko, Phys. Lett. B, 251 (1990), 517 | DOI | MR

[34] A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov, A. Orlov, Nucl. Phys. B, 357 (1991), 565 | DOI | MR

[35] H. Itoyama, Y. Matsuo, Phys. Lett. B, 255 (1991), 202 | DOI | MR

[36] H. S. La, Commun. Math. Phys., 140 (1991), 569 | DOI | MR | Zbl

[37] Yu. Makeenko, A. Marshakov, A. Mironov, A. Morozov, Nucl. Phys. B, 356 (1991), 574 | DOI | MR

[38] C. Itzykson, J.-B. Zuber, Int. J. Mod. Phys. A, 7 (1992), 5661 | DOI | MR | Zbl

[39] R. Dijkgraaf, E. Verlinde, H. Verlinde, Nucl. Phys. B, 348 (1991), 435 | DOI | MR

[40] D. J. Gross, M. J. Newman, Phys. Lett. B, 266 (1991), 291 ; Nucl. Phys. B, 380 (1992), 168 | DOI | MR | DOI | MR

[41] A. Marshakov, A. Mironov, A. Morozov, Phys. Lett. B, 274 (1992), 280 | DOI | MR

[42] R. Dijkgraaf, E. Verlinde, H. Verlinde, Nucl. Phys. B, 352 (1991), 59 | DOI | MR

[43] E. Witten, Nucl. Phys. B, 371 (1992), 191 | DOI | MR

[44] S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, A. Zabrodin, Phys. Lett. B, 275 (1992), 311 | DOI | MR

[45] E. Brézin, D. Gross, Phys. Lett. B, 97 (1980), 120 | DOI | MR

[46] R. C. Brower, M. Nauenberg, Nucl. Phys. B, 180[FS2] (1981), 221 | DOI | MR

[47] Yu. Makeenko, G. Semenoff, Mod. Phys. Lett. A, 6 (1991), 3455 | DOI | MR | Zbl

[48] Yu. Makeenko, Mod. Phys. Lett. (Brief Reviews) A, 6 (1991), 1901 | DOI | MR | Zbl

[49] V. A. Voevodskii, G. B. Shabat, DAN SSSR, 304:2 (1989), 265 | MR | Zbl

[50] G. Belyi, Izv. AN SSSR. Ser. matem., 43:2 (1979), 267 | MR | Zbl

[51] P. B. Cohen, C. Itzykson, J. Wolfart, Preprint Saclay T93/048; Commun. Math. Phys., 163 (1994), 605 | DOI | MR | Zbl

[52] L. Chekhov, Geom. and Phys., 12 (1993), 153 | DOI | MR | Zbl

[53] J. Ambjørn, L. Chekhov, C. F. Kristjansen, Yu. Makeenko, Nucl. Phys. B, 404 (1993), 127 | DOI | MR

[54] L. Chekhov, Yu. Makeenko, Mod. Phys. Lett. A, 7 (1992), 1223 | DOI | MR | Zbl

[55] J. Harer, D. Zagier, Invent. Math., 85 (1986), 457 | DOI | MR | Zbl

[56] R. C. Penner, Commun. Math. Phys., 113 (1987), 299 ; J. Diff. Geom., 27 (1988), 35 | DOI | MR | Zbl | DOI | MR | Zbl

[57] J. Distler, C. Vafa, Mod. Phys. Lett. A, 6 (1991), 259 | DOI | MR | Zbl

[58] L. Chekhov, A. Zabrodin, Mod. Phys. Lett. A, 6 (1991), 3143 | DOI | MR | Zbl

[59] L. O. Chekhov, Matrichnye modeli i geometriya prostranstva modulei, Diss. ...dokt. f.-m. n., MIAN, M., 2000

[60] J. Ambjørn, L. Chekhov, Yu. Makeenko, Phys. Lett. B, 282 (1992), 341 | DOI | MR

[61] D. Gross, E. Witten, Phys. Rev. D, 21 (1980), 446 | DOI

[62] L. Chekhov, Yu. Makeenko, Phys. Lett. B, 278 (1992), 271 | DOI | MR

[63] A. Fayyazuddin, Y. Makeenko, P. Olesen, D. J. Smith, K. Zarembo, Nucl. Phys. B, 499 (1997), 159 ; E-print hep-th/9703038 | DOI | MR | Zbl | MR

[64] C. F. Kristiansen, P. Olesen, Phys. Lett. B, 405 (1997), 45 ; E-print hep-th/9704017 | DOI | MR

[65] A. Mironov, A. Morozov, G. W. Semenoff, Int. J. Mod. Phys. A, 11 (1996), 5031 | DOI | MR | Zbl

[66] L. Chekhov, K. Zarembo, Mod. Phys. Lett. A, 12 (1997), 2331 | DOI | MR | Zbl

[67] J. Ambjørn, L. Chekhov, The NBI matrix model of IIB superstring, Preprint JHEP 9812:007, 1998 ; E-print hep-th/9805212 | MR

[68] L. Chekhov, K. Palamarchuk, Mod. Phys. Lett. A, 14:32 (1999), 2229 | DOI | MR

[69] I. Kostov, Mod. Phys. Lett. A, 4 (1989), 217 | DOI | MR

[70] B. Eynard, J. Zinn-Justin, Nucl. Phys. B, 386 (1992), 558 | DOI | MR

[71] B. Eynard, C. F. Kristjansen, Nucl. Phys. B, 455 (1995), 577 ; E-print hep-th/9506193 | DOI | MR | Zbl

[72] L. Chekhov, C. Kristjansen, Nucl. Phys. B, 479[FS] (1996), 683 | DOI | MR | Zbl

[73] K. Ueno, K. Takasaki, “Toda lattice hierarchy”, Group Representation and Systems of Differential Equations (Tokyo, 1982), Adv. Stud. Pure Math., 4, ed. K. Okamoto, North-Holland, Amsterdam, 1984, 1 | MR

[74] K. Strebel, Quadratic Differentials, Springer, Berlin, 1984 | MR

[75] J. Harer, “The cohomology of the moduli space of curves”, Theory of Moduli (Montecatini Terme, 1985), Lect. Notes in Math., 1337, ed. S. Sernesi, Springer, Berlin, 1988, 138 | DOI | MR

[76] P. Deligne, M. Mumford, Publ. IHES, 45 (1969), 75 | DOI | MR

[77] D. Mumford, “Towards an enumerative geometry of the moduli space of curves”, Arithmetic and Geometry, II, Progr. Math., 36, eds. M. Artin, J. Tate, Birkhäuser, Boston, MA, 1983, 271 | MR

[78] R. Dijkgraaf, “Intersection theory, integrable hierarchies and topological field theory”, New Symmetry Principles in Quantum Field Theory, Proc. of NATO Adv. Study Inst. (Cargese, July 16–27, 1991), eds. J. Frölich, G. 't Hooft, A. Jaffe, G. Mack, P. K. Mitter, R. Stora, Plenum, New York, 1992, 95 ; E-print hep-th/9201003 | DOI | MR | Zbl

[79] V. Kazakov, Mod. Phys. Lett. A, 4 (1989), 2125 | DOI | MR

[80] A. A. Migdal, Phys. Rep., 102 (1983), 199 | DOI

[81] S. Dalley, C. Johnson, T. Morris, Nucl. Phys. B, 368 (1992), 625 ; 655 | DOI | MR | MR

[82] K. L. Zarembo, L. O. Chekhov, TMF, 93:2 (1992), 354 | MR | Zbl

[83] F. David, Mod. Phys. Lett. A, 5 (1990), 1019 | DOI | MR | Zbl

[84] J. Ambjørn, C. F. Kristjansen, Yu. Makeenko, Mod. Phys. Lett. A, 7 (1992), 3187 ; E-print hep-th/9207020 | DOI | MR | Zbl

[85] L. Alvarez-Gaumé, J. L. Mañes, Mod. Phys. Lett. A, 6 (1991), 2039 ; L. Alvarez-Gaumé, H. Itoyama, J. L. Mañes, A. Zadra, Int. J. Mod. Phys. A, 7 (1992), 5337 ; E-print hep-th/9112018 | DOI | MR | Zbl | DOI | MR | Zbl

[86] J. C. Plefka, Nucl. Phys. B, 448 (1995), 355 ; E-print hep-th/9504089 | DOI

[87] J. Ambjørn, Yu. Makeenko, Mod. Phys. Lett. A, 5 (1990), 1753 | DOI | MR | Zbl

[88] C. Bachas, P. M. S. Petropoulos, Phys. Lett. B, 247 (1990), 363 | DOI | MR

[89] Yu. Makeenko, “Loop equations and Virasoro constraints in matrix models”, Proc. XXVth Intl. Symp. on the Theory of Elementary Particles, ed. H. J. Kaiser, Gösen, 1991, 275

[90] L. Motl, Proposals on nonperturbative superstring interactions, E-print hep-th/9701025

[91] S. Sethi, L. Susskind, Phys. Lett. B, 400 (1997), 265 ; E-print hep-th/9702101 | DOI | MR

[92] T. Banks, N. Seiberg, Nucl. Phys. B, 497 (1997), 41 ; E-print hep-th/9702187 | DOI | MR | Zbl

[93] R. Dijkgraaf, E. Verlinde, H. Verlinde, Nucl. Phys. B, 500 (1997), 43 ; E-print hep-th/9703030 | DOI | MR | Zbl

[94] M. Grin, Dzh. Shvarts, E. Vitten, Vedenie v teoriyu superstrun, Mir, M., 1990 | MR

[95] J. Polchinsky, Superstrings, V. 1, 2, Cambridge Univ. Press, Cambridge, 1998 | MR

[96] T. Yoneya, Progr. Theor. Phys., 97 (1997), 949 ; E-print hep-th/9703078 | DOI

[97] K. L. Zarembo, Yu. M. Makeenko, UFN, 168:1 (1998), 3 | DOI

[98] L. Paniak, N. Weiss, J. Math. Phys., 36 (1995), 2512 ; E-print hep-th/9501037 | DOI | MR | Zbl

[99] R. Dijkgraaf, G. Moore, R. Plesser, Nucl. Phys. B, 394 (1993), 356 | DOI | MR

[100] C. Imbimbo, S. Mukhi, Nucl. Phys. B, 449 (1995), 553 | DOI | MR | Zbl

[101] R. C. Brower, P. Rossi, C.-I. Tan, Phys. Rev. D, 23 (1981), 942 | DOI

[102] Yu. Makeenko, Phys. Lett. B, 314 (1993), 197 | DOI | MR

[103] Yu. Makeenko, Int. J. Mod. Phys. A, 10 (1995), 2615 | DOI | MR | Zbl

[104] L. Chekhov, Acta Appl. Math., 48 (1997), 33 | DOI | MR | Zbl

Parcourir par

Geodesic

Parcourir par