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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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