Geodesic
Generalized integrability and two-dimensional gravitation
Teoretičeskaâ i matematičeskaâ fizika, Tome 95 (1993) no. 2, pp. 258-275 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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T. Hollowood; J. L. Miramontes; J. S. Guillén. Generalized integrability and two-dimensional gravitation. Teoretičeskaâ i matematičeskaâ fizika, Tome 95 (1993) no. 2, pp. 258-275. http://geodesic.mathdoc.fr/item/TMF_1993_95_2_a10/

[1] David F., Nucl. Phys., B257 (1985), 45 ; Kazakov V. A., Kostov I. A., Migdal A. A., Phys. Lett., B157 (1985), 295 ; Kostov I. K., Mehta M. L., Phys. Lett., B189 (1987), 118 | DOI | MR | DOI | MR | DOI | MR

[2] Douglas M., Shenker S., Nucl. Phys., B355 (1990), 635 | DOI | MR | Zbl

[3] Douglas M., Phys. Lett., B238 (1990), 176 | DOI | MR | Zbl

[4] Drinfel'd V. G., Sokolov V. V., Jour. Sov. Math., 30 (1985), 1975 ; Soviet. Math. Dokl., 23 (1981), 457 | DOI | Zbl | Zbl

[5] Brézin E., Douglas M., Kazakov V., Shenker S., Phys. Lett., 237B (1990), 43 | DOI | MR

[6] Di Francesco P., Kutasov D., Nucl. Phys., B342 (1990), 589 | DOI

[7] Kazama Y., Suzuki H., Phys. Lett., 216B (1989), 112 | DOI | MR

[8] Dijkgraaf R., Verlinde H., Verlinde E., Nucl. Phys., B352 (1991), 59 ; Yen T., Phys. Lett., B252 (1990), 361 | DOI | MR | DOI | MR

[9] Kontsevich M., Commun. Math. Phys., 147 (1992), 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., B380 (1992), 168 ; Kharchev S., Marshakov A., Mironov A., Morozov A., Zabrodin A., Nucl. Phys., B380 (1992), 181 | DOI | MR | Zbl | MR | DOI | MR | DOI | MR

[10] Kac V. G., Wakimoto M., Proceedings of Symposia in Pure Mathematics, 49 (1989), 191 | DOI | MR | Zbl

[11] De Groot M. F., Hollowood T. J., Miramontes J. L., Commun. Math. Phys., 145 (1992), 57 | DOI | MR | Zbl

[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., B277 (1992), 89 | MR | DOI | MR

[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 | MR

[14] Burroughs N. J., Nucl. Phys., B379 (1992), 340 | DOI | MR

[15] Kac V. G., Infinite dimensional Lie algebras, 3-rd edition, Cambridge University Press, 1990 | MR

[16] Kac V. G., Peterson D. H., “$112$ constructions of the basic representation of the loop group of $\operatorname{E}_8$”, Symposium on anomalies, geometry, topology (Chicago, Ill., 1985), World Sci. Publishing, Singapore, 1985, 276–298 | MR

[17] Witten E., Nucl. Phys., B340 (1990), 281 ; Distler J., Nucl. Phys., B342 (1990), 523 ; Verlinde E., Verlinde H., Nucl. Phys., B348 (1991), 457 ; Li K., Nucl. Phys., B354 (1991), 711 ; B354 (1991), 725 ; Dijkgraaf R., Intersection theory, Integrable hierarchies and Topological field theory, preprint IASSNS-HEP-91/91 | DOI | MR | DOI | MR | DOI | MR | MR

[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., B357 (1991), 632 ; Fehér L., O'Raifeartaigh L., Ruelle P., Tsutsui I., Wipf A., Ann. Phys. (N. Y.), 213 (1992), 1 ; “On Hamiltonian reduction of the WZNW theories”, Phys. Rep., 222:1 (1992), 64 | MR | DOI | MR | DOI | MR | Zbl | DOI | MR

[19] Polyakov A., Int. J. Mod. Phys., A5 (1990), 833 ; Berhadsky M., Commun. Math. Phys., 139 (1991), 71 ; Mathieu P., Oevel W., Mod. Phys. Lett., A6 (1991), 2397 | DOI | MR | DOI | MR | DOI | MR | Zbl

[20] Bakas I., Depireux D. A., Mod. Phys. Lett., A6 (1991), 1561 ; Erratum, A6 (1991), 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 | DOI | MR | Zbl | DOI | MR | Zbl | MR | MR

[21] Hirota R., “Direct methods in soliton theory”, Soliton, eds. Bullogh R. K., Caudrey P. S., 1980, 157 | DOI | MR

[22] Peterson D. H., Kac V. G., Proc. Nat. Acad. Sci. USA, 80 (1983), 1778 | DOI | MR | Zbl

[23] Imbens H-J., “Drinfel'd–Sokolov hierarchies and $\tau$-functions”, Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys., 7, World Sci. Publ., 1989, 352–368 | MR

[24] Alvarez-Gaumé L., Gómez C., Lacki J., Phys. Lett., 253B (1991), 56 ; Gerasimov A., Marshakov A., Mironov A., Morozov A., Orlov A., Nucl. Phys. 1991, B357, 565 ; Martinec E., Commun. Math. Phys., 138 (1991), 437 | DOI | MR | MR | DOI | MR | Zbl

[25] Crnković C., Moore G., Phys. Lett., B247 (1991), 322 | DOI | MR

[26] Moore G., Commun. Math. Phys., 133 (1990), 261 | DOI | MR | Zbl

[27] Orlov A. Y., Schulman E. I., Lett. in Math. Phys., 12 (1986), 171 ; Grinevich P. G., Orlov A. Y., “Virasoro action on Riemann surfaces, Grassmanians, $\det\overline{\partial}_j$, and Segal-Wilson $\tau$-function”, 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”, Nonlinear evolution equations and dynamical systems, eds. Carrillo S., Ragnisco O., Springer, 1990 | DOI | MR | Zbl | MR | MR

[28] Semikhatov A. M., Virasoro action and Virasoro constraints on integrable hierarchies of the $r$-matrix type, P. N. Lebedev Phys. Inst. preprint hep-th/9112016 ; Nucl. Phys., B366 (1991), 347 | MR | DOI | MR

[29] Periwal V., Shevitz D., Phys. Rev. Lett., 64 (1990), 1326 ; Nucl. Phys., B344 (1990), 731 ; Demeterfi K., Tan C-J., Mod. Phys. Lett., A5 (1990), 1563 ; Myers R. C., Periwal V., Phys. Rev. Lett., 65 (1990), 1088 ; Bowick M. J., Morozov A., Shevitz D., Nucl. Phys., B354 (1991), 496 | DOI | MR | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR

[30] Crnković C., Douglas M., Moore G., Nucl. Phys., B360 (1991), 507 | DOI | MR

[31] Dalley S., Johnson C. V., Morris T. M., Wätterstam A., Princeton Univ. preprint PUPT-1325 (hep-th/9206060) | MR

[32] Dijkgraaf R., Verlinde H., Verlinde E., Nucl. Phys., B384 (1991), 435 ; Fukuma M., Kawai H., Nakayama R., Int. Jour. Mod. Phys., A6 (1991), 1385 ; Commun. Math. Phys., 143 (1992), 371 ; Goeree J., Nucl. Phys., B358 (1991), 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) | DOI | MR | DOI | MR | DOI | MR | Zbl | DOI | MR | MR

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

[34] Pasquinucci A., A note on the Zakharov–Shabat topological model, Princeton Univ. preprint PUPT-1308 (hep-th/9203028) | MR

[35] Hollowood T., Miramontes J. L., Pasquinucci A., Nappi C., Nucl. Phys., B373 (1992), 247 | DOI | MR