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Generalized integrability and two-dimensional gravitation

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Abstract

We review the construction of generalized integrable hierarchies of partial differential equations, associated to affine Kac-Moody algebras, that include those considered by Drinfel'd and Sokolov. These hierarchies can be used to construct new models of 2D quantum or topological gravity, as well as new W-algebras.

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References

  1. David F. // Nucl. Phys. 1985. V. B257. P. 45;Kazakov V.A., Kostov I.A., Migdal A.A. // Phys. Lett. 1985. V. B157. P. 295;Kostov I.K., Mehta M.L. // Phys. Lett. 1987. V. B189. P. 118.

    Google Scholar 

  2. Douglas M., Shenker S. // Nucl. Phys. 1990. V. B355. P. 635;Gross D., Migdal A.A. // Nucl. Phys. 1990. V. B340. P. 333;Brézin E., Kazakov V.A. // Phys. Lett. 1990. V. B236. P. 144.

    Google Scholar 

  3. Douglas M. // Phys. Lett. 1990. V. B238. P. 176.

    Google Scholar 

  4. Drinfel'd V.G., Sokolov V.V. // Jour. Sov. Math. 1985. V. 30. P. 1975; Soviet. Math. Dokl. 1981. V. 23. P. 457.

    Google Scholar 

  5. Brezin E., Douglas M., Kazakov V., Shenker S. // Phys. Lett. 1990. V. 237B. P. 43;Gross D., Migdal A. // Phys. Rev. Lett. 1990. V. 64. P. 717;Crnković C., Ginsparg P., Moore G. // Phys. Lett. 1990. V. 237B. P. 196.

    Google Scholar 

  6. Di Francesco P., Kutasov D. // Nucl. Phys. 1990. V. B342. P. 589.

    Google Scholar 

  7. Kazama Y., Suzuki H. // Phys. Lett. 1989. V. 216B. P. 112; Nucl. Phys. 1989. V. B321. P. 232.

    Google Scholar 

  8. Dijkgraaf R., Verlinde H., Verlinde E. // Nucl. Phys. 1991. V. B352. P. 59;Yen T. // Phys. Lett. 1990. V. B252. P. 361.

    Google Scholar 

  9. Kontsevich M. // Commun. Math. Phys. 1992. V. 147. P. 1;Witten E. On the Kontsevitch model and other models of two dimensional gravity: preprint IASSNS-HEP-91/24 (1991);Gross D.J., Newman M.J. // Nucl. Phys. 1992. V. B380. P. 168;Kharchev S., Marshakov A., Mironov A., Morozov A., Zabrodin A. // Nucl. Phys. 1992. V. B380. P. 181.

    Google Scholar 

  10. Kac V.G., Wakimoto M. // Proceedings of Symposia in Pure Mathematics. 1989. V. 49. P. 191.

    Google Scholar 

  11. De Groot M.F., Hollowood T.J., Miramontes J.L. // Commun. Math. Phys. 1992. V. 145. P. 57.

    Google Scholar 

  12. Burroughs N.J., de Groot M.F., Hollowood T.J., Miramontes J.L. Generalized Drinfel'd-Sokolov hierarchies II: the Hamiltonian structures. Princeton Preprint PUPT-1263, IASSNS-HEP-91/42; // Phys. Lett. 1992. V. B277. P. 89.

  13. Hollowood T.J., Miramontes J.L. Tau-functions and generalized integrable hierarchies. CERN and Oxford Univ. Preprint, CERN-TH-6594/92, OUTP-92-15P.

  14. Burroughs N.J. // Nucl. Phys. 1992. V. B379. P. 340; A loop algebra coadjoint orbit construction of the generalized KdV hierarchies. preprint IASSNS-HEP-91-95, DAMTP-92-21 (1992);van Driel P. // Phys. Lett. 1992. V. B274. P. 179;Fehér L., Harnad J., Marshall I. Generalized Drinfel'd-Sokolov reductions and KdV type hierarchies. Univ. de Montreal preprint UdeM-LPN-TH-92/103, CRM-1832 (1992) (hep-th/9210037).

    Google Scholar 

  15. Kac V.G. Infinite dimensional Lie algebras. 3rd edition, Cambridge University Press, 1990.

  16. Kac V.G., Peterson D.H.. 112 constructions of the basic representation of the loop group of Es. In: Symp. on anomalies, geometry and topology. Eds. Bardeen W.A., White A.R.. Singapore: World Scientific, 1985.

    Google Scholar 

  17. Witten E. // Nucl. Phys. 1990. V. B340. P. 281;Distler J. // Nucl. Phys. 1990. V. B342 P. 523;Verlinde E., Verlinde H. // Nucl. Phys. 1991. V. B348. P. 457;Li K. // Nucl. Phys. 1991. V. B354. P. 711; ibid. Nucl. Phys. 1991. V. B354. P. 725; for a general review seeDijkgraaf R. Intersection theory, Integrable hierarchies and Topological field theory. Preprint IASSNS-HEP-91/91, and references therein.

    Google Scholar 

  18. Frappat L., Ragoucy E., Sorba P. W-algebras and superalgebras from constrained WZW models: a group theoretical classification. Preprint ENSLAPP-AL-391/92 (1992);Bais F.A., Tjin T., van Driel P. // Nucl. Phys. 1991. V. B357. P. 632;Fehér L., O'Raifeartaigh L., Ruelle P., Tsulsui, I., Wipf A. // Ann. Phys. (N.Y.) 1992. V. 213. P. 1; On Hamiltonian reduction of the WZNW theories, to appear in Phys. Rep.

  19. Polyakov A. // Int. J. Mod. Phys. 1990. V. A5. P. 833;Berhadsky M. // Commun. Math. Phys. 1991. V. 139. P. 71;Mathieu P., Oevel W. // Mod. Phys. Lett. 1991. V. A6. P. 2397.

    Google Scholar 

  20. Bakas I., Depireux D.A. // Mod. Phys. Lett. 1991. V. A6. P. 1561 (Erratum 1991. V. A6. P. 2351); The origin of gauge symmetries in integrable systems of KdV type: Univ. Of Maryland preprint UMD-PP91-111 (1990); Selfduality, KdV flows and W-algebras: Univ. of Maryland preprint PRINT-91-0426, C91-06-03.3 (1991).

    Google Scholar 

  21. Hirota R. Direct methods in soliton theory. In: Soliton. Eds. Bullogh R.K., Caudrey P.S. 1980. P. 157.

  22. Peterson D.H., Kac V.G. // Proc. Nat. Acad. Sci. U.S.A. 1983. V. 80. P. 1778.

    Google Scholar 

  23. Imbens H-J. Drinfel'd-Sokolv hierarchies and τ-functions. In: Infinite dimensional Lie algebras and groups. World Scientific. Adv. Ser. in Math. Phys. 1989. V. 7. P. 352.

  24. Alvarez-Gaumé L., Gómez C., Lacki J. // Phys. Lett. 1991. V. 253B. P. 56;Gerasimov A., Marshakov A., Mironou A., Morozov A., Orlov A. // Nucl. Phys. 1991. V. B357. P. 565;Martinec E. // Commun. Math. Phys. 1991. V. 138. P. 437.

    Google Scholar 

  25. Crnković C., Moore G. // Phys. Lett. 1991. V. B247. P. 322.

    Google Scholar 

  26. Moore G. // Commun. Math. Phys. 1990. V. 133. P. 261; Prog. Theor. Phys. Suppl. 1990. V. 102. P. 255.

    Google Scholar 

  27. Orlov A.Y., Schulman E.I. // Lett. in Math. Phys. 1986. V. 12. P. 171;Grinevich P.G., Orlov A.Y. Virasoro action on Riemann surfaces, Grassmanians, det. 566-1, and Segal-Wilson τ-function. In: Problems of Modern Quantum Field Theory. Eds. Belavin A.A., Klimyk A.U., Zamolodchikov A.B. Springer, 1989; Higher symmetries of KP equation and the Virasoro action on Riemann surfaces. In: Nonlinear evolution equations and dynamical systems. Eds. Carrillo S., Ragnisco O. Springer, 1990.

    Google Scholar 

  28. Semikhatov A.M. Virasoro action and Virasoro constraints on integrable hierarchies of ther-matrix type. P.N. Lebedev Phys. Inst. Preprint hep-th/9112016; // Nucl. Phys. 1991. V. B366. P. 347.

    Google Scholar 

  29. Periwal V., Shevitz D. // Phys. Rev. Lett. 1990. V. 64. P. 1326; Nucl. Phys. 1990. V. B344. P. 731;Demeterfi K., Tan C.-I. // Mod. Phys. Lett. 1990. V. A5. P. 1563;Myers R.C., Periwal V. // Phys. Rev. Lett. 1990. V. 65 P. 1088;Bowick M.J., Morozov A., Shevitz D. // Nucl. Phys. 1991. V. B354. P. 496.

    Google Scholar 

  30. Crnković C., Douglas M., Moore G. // Nucl. Phys. 1991. V. B360. P. 507.

    Google Scholar 

  31. Dalley. S., Johnson C.V., Morris T.M. Wätterstam A. Princeton Univ. preprint PUPT-1325 (hepth/9206060) and references therein.

  32. Dijkgraaf R., Verlinde H., Verlinde E. // Nucl. Phys. 1991. V. B384. P. 435;Fukuma M., Kawai H., Nakayama R. // Int. Jour. Mod. Phys. 1991. V. A6. P. 1385; Commun. Math. Phys. 1992. V. 143. P. 371;Goeree J. // Nucl. Phys. 1991. V. B358. P. 737;Panda S., Roy S. The Lax operator approach for the Virasoro and the W-constraints in the generalized KdV hierarchy. ICTP preprint IC-92-226 (hep-th/9208065).

    Google Scholar 

  33. Fateev V.A., Zamolodchikov A.B. // Nucl. Phys. 1987. V. B280 [FS18].P. 644;Fateev V.A., Lykyanov S.L. // Int. J. Mod. Phys. 1988. V. A3. P. 507;Bais F.A., Bouwknegt P., Surridge M., Schoutens K. // Nucl. Phys. 1988. V. B304. P. 348; ibid. Nucl. Phys. 1988. V. B304. P. 371.

    Google Scholar 

  34. Pasquinucci A. A note on the Zakharov-Shabat topological model. Princeton Univ. preprint PUPT-1308 (hep-th/9203028);Bonora L., Xiong C.S. Matrix models without scaling limit. SISSA/ISAS preprint SISSA-ISAS 161/92/EP (hep-th/9209041).

  35. Hollowood T., Miramontes J.L., Pasquinucci A., Nappi C. // Nucl. Phys. 1992. V. B373. P. 247.

    Google Scholar 

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Authors
  1. Timothy Hollowood
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  2. J. Luis Miramontes
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  3. Joaquín Sánchez Guillén
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Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 2, pp. 258–275, May, 1993.

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Hollowood, T., Miramontes, J.L. & Guillén, J.S. Generalized integrability and two-dimensional gravitation. Theor Math Phys 95, 552–567 (1993). https://doi.org/10.1007/BF01017141

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