Communications in Mathematical Physics

, Volume 145, Issue 2, pp 329–344 | Cite as

String vertices, overlap equations, τ functions and the Hirota equation

  • B. E. W. Nilsson
  • P. West


String vertices,V, are shown to satisfy a new type of overlap equation of the form\(V \exp \{ ip \cdot Q^i (\xi ^i )\} = V \exp \{ ip \cdot Q^i (\xi ^i )\} \left( {\frac{{d\xi ^j }}{{d\xi ^i }}} \right)^{p^2 /2} \) as well as corresponding equations forAn andBn cycles. A special case of such an equation, when integrated, is shown to be the Hirota equation for the K−P hierarchy.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    For a review see: Alessandrini, V., Amati, D., Le Bellac, M., Olive, D.: Phys. Rep.C 1, 269 (1971)CrossRefGoogle Scholar
  2. 2.
    Kaku, M., Kikkawa, K.: Phys. Rev.D10, 110, 1823 (1974)CrossRefGoogle Scholar
  3. 3.
    For a review, see Hoker, E. D., Phong, D.: The geometry of string peturbation theory. Rev. Mod. Phys.Google Scholar
  4. 4.
    Neveu, A., West, P.: Phys. Lett.168B, 192 (1985); Nucl. Phys.B278, 601 (1986),B293, 266 (1987); Witten, E.: Nucl. Phys.B268, 253 (1986); Hata, H., Itoh, K., Kugo, T., Ogawa, K.: Phys. Lett.172B, 186 (1986)Google Scholar
  5. 5.
    Neveu, A., West, P.: Nucl. Phys.B278, 601 (1986) Sect. 3CrossRefGoogle Scholar
  6. 6.
    Neveu, A., West, P.: Phys. Lett.B179, 235 (1986),B183, 34 (1986);B198, 187 (1987); Kazama, Y., Neveu, A., Nicolai, H., West, P.: Nucl. Phys.B278, 832 (1986)CrossRefGoogle Scholar
  7. 7.
    Neveu, A., West, P.: Phys. Lett.B193, 187 (1987),B194, 200 (1987),B200, 275 (1988); Commun. Math. Phys.114, 613 (1988),119, 585 (1988); West, P.: Phys. Lett.B205, 38 (1988); Freeman, M., West, P.: Phys. Lett.B205, 30 (1988)CrossRefGoogle Scholar
  8. 8.
    For a review, see: West, P.: A brief review of the group theoretic approach to string theory, in: Conformal field theories and related topics. Binétrug, P., Sorba, P., Stora, R. (eds.) Nucl. Phys. B. Proc. Suppl.5B, 217 (1988)Google Scholar
  9. 9.
    West, P.: Nucl. Phys.B320, 103 (1989); Introduction to string theory, Zakopane, 1988; Acta Phys. PolanicaB20, 471 (1989) and In: Superstrings, Unfied Theories and Cosmology 1988, Ellis, G., Pati, J., Randjbar-Daemi, S., Sezgin, E., Shafi, Q. (eds.). Singapore: World Scientific 1989CrossRefGoogle Scholar
  10. 10.
    West, P.: to appearGoogle Scholar
  11. 11.
    Mandelstam, S.: Workshop on unified string theories. Green, M., Gross, D. (eds.), Singapore: World Scientific 1985; Ford, L.: Automorphic functions. New York: Chelsea 1951; Burnside, W.: Proc. Lond. Math. Soc.23, 49 (1981); Alessandrini, V.: Nuovo Cim.2A, 321 (1971),4A, 793 (1971)Google Scholar
  12. 12.
    Dijkgraaf, R., Verlinde, E., Verlinde, H.: Commun. Math. Phys.115, 649 (1988)CrossRefGoogle Scholar
  13. 13.
    Di Vecchia, P., Pezzella, F., Frau, M., Hornfeck, K., Levela, A., Sciuto, A.: Nucl. Phys.B322, 317 (1989)CrossRefGoogle Scholar
  14. 14.
    For a review, see Kac, V., Raina, A.: Highest weight representations of infinite dimensional Lie algebras. Singapore: World Scientific 1987; Date, E., Jimbo, M., Kashivara, M., Miwa, T.: Transformation groups for soliton equation. In: Non-linear integrable systems, Jimbo, M., Miwa, T. (eds.), Singapore: World Scientific 1983; Shiota, T.: Invent. Math.83, 333 (1986); Mulase, M.: J. Diff. Geom.19, 403 (1984) and references thereinGoogle Scholar
  15. 15.
    Saito, S.: Phys. Rev.D37, 990 (1988); Characterization of string amplitudes by discrete equations. Takyo preprint TMU P-HEL-8812Google Scholar
  16. 16.
    Ishibashi, N., Matsu, Y., Ooguri, H.: Mod. Phys. Lett.A2, 119 (1987)CrossRefGoogle Scholar
  17. 17.
    West, P.: Gauge covariant and dual model vertices. In: Frontiers of High Energy Physics. Halliday, I. (ed.). London: Adam Hilger 1987Google Scholar
  18. 18.
    Di Vechia, P., Nakayama, R., Petersen, J. L., Sidenius, J., Sciuto, S.: Nucl. Phys.B287, 621 (1987); Petersen, J. L., Roland, K. O., Sidenius, J. R.: Phys. Lett.B205, 262 (1988)CrossRefGoogle Scholar
  19. 19.
    Di Vecchia, P., Frau, M., Lerda, A., Sciuto, S.: Phys. Lett.B199, 49 (1987);B206, 643 (1988); Di Vecchia, P., Hornfeck, K., Frau, M., Lerda, A., Sciuto, S.: In: Perspectives in String Theory, Di Vecchia, P., Petersen (eds.), p. 422. Singapore: World Scientific 1988CrossRefGoogle Scholar
  20. 20.
    Alvarez-Gaumé, L., Gomez, C., Reina, C.: Phys. Lett.B190, 55 (1987); Alvarez-Gaumé, L., Gomez, C., Moore, G., Vafa, C.: Nucl. Phys.B303, 455 (1988)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • B. E. W. Nilsson
    • 1
  • P. West
    • 2
  1. 1.Institute of Theoretical PhysicsChalmers University of TechnologyGoteborgSweden
  2. 2.Mathematics DepartmentKing's College LondonLondon

Personalised recommendations