Letters in Mathematical Physics

, Volume 104, Issue 11, pp 1407–1423 | Cite as

Quantum Torus Symmetry of the KP, KdV and BKP Hierarchies

  • Chuanzhong Li
  • Jingsong HeEmail author


In this paper, we construct the quantum torus symmetry of the KP hierarchy and further derive the quantum torus constraint on the tau function of the KP hierarchy. That means we give a nice representation of the quantum torus Lie algebra in the KP system by acting on its tau function. Comparing to the W symmetry, this quantum torus symmetry has a nice algebraic structure with double indices. Further by reduction, we also construct the quantum torus symmetries of the KdV and BKP hierarchies and further derive the quantum torus constraints on their tau functions. These quantum torus constraints might have applications in the quantum field theory, supersymmetric gauge theory and so on.


KP hierarchy quantum torus symmetry quantum torus constraint KdV hierarchy BKP hierarchy 

Mathematics Subject Classification

37K05 37K10 37K40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dickey L.A.: Soliton Equations and Hamiltonian Systems, 2nd edn. World Scientific, Singapore (2003)CrossRefzbMATHGoogle Scholar
  2. 2.
    Orlov A.Yu., Schulman E.I.: Additional symmetries of integrable equations and conformal algebra representation. Lett. Math. Phys. 12, 171–179 (1986)MathSciNetCrossRefzbMATHADSGoogle Scholar
  3. 3.
    He J.S., Tian K.L., Forester A., Ma W.X.: Additional symmetries and string equation of the CKP hierarchy. Lett. Math. Phys. 81, 119–134 (2007)MathSciNetCrossRefzbMATHADSGoogle Scholar
  4. 4.
    Tian K.L., He J.S., Cheng J.P., Cheng Y.: Additional symmetries of constrained CKP and BKP hierarchies. Sci. China Math. 54, 257–268 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Li M.H., Li C.Z. et al.: Virasoro type algebraic structure hidden in the constrained discrete Kadomtsev–Petviashvili hierarchy. J. Math. Phys. 54, 043512 (2013)MathSciNetCrossRefADSGoogle Scholar
  6. 6.
    Block R.: On torsion-free abelian groups and Lie algebras. Proc. Am. Math. Soc. 9, 613–620 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dokovic D., Zhao K.: Derivations, isomorphisms and second cohomology of generalized Block algebras. Algebra Colloq. 3, 245–272 (1996)MathSciNetGoogle Scholar
  8. 8.
    Su Y.: Quasifinite representations of a Lie algebra of Block type. J. Algebra 276, 117–128 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, M.R., et al.: W 1+∞ 3-algebra and KP hierarchy. arXiv:1309.4627
  10. 10.
    Kemmoku R., Saito S.: Discretization of Virasoro algebra. Phys. Lett. B 319, 471–477 (1993)MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Mas J., Seco M.: The algebra of q-pseudodifferential symbols and the \({q-W_{KP}^{(n)}}\) algebra. J. Math. Phys. 37, 6510 (1996)MathSciNetCrossRefzbMATHADSGoogle Scholar
  12. 12.
    Li C.Z., He J.S., Su Y.C.: Block type symmetry of bigraded Toda hierarchy. J. Math. Phys. 53, 013517 (2012)MathSciNetCrossRefADSGoogle Scholar
  13. 13.
    Li C.Z., He J.S., Wu K., Cheng Y.: Tau function and Hirota bilinear equations for the extended bigraded Toda hierarchy. J. Math. Phys. 51, 043514 (2010)MathSciNetCrossRefADSGoogle Scholar
  14. 14.
    Li C.Z.: Solutions of bigraded Toda hierarchy. J. Phys. A 44, 255201 (2011)MathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Li C.Z., He J.S.: Dispersionless bigraded Toda hierarchy and its additional symmetry. Rev. Math. Phys. 24, 1230003 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Li, C.Z., He, J.S.: Block algebra in two-component BKP and D type Drinfeld–Sokolov hierarchies. J. Math. Phys. 54, 113501 (2013). arXiv:1210.6498
  17. 17.
    Fairlie D.B., Fletcher P., Zachos C.K.: Trigonometric structure constants for new infinite-dimensional algebras. Phys. Lett. B 218, 203 (1989)MathSciNetCrossRefzbMATHADSGoogle Scholar
  18. 18.
    Maeda T., Nakatsu T., Takasaki K., Tamakoshi T.: Free fermion and Seiberg–Witten differential in random plane partitions. Nucl. Phys. B 715, 275 (2005)MathSciNetCrossRefzbMATHADSGoogle Scholar
  19. 19.
    Maeda T., Nakatsu T.: Amoebas and instantons. Int. J. Mod. Phys. A 22, 937 (2007)MathSciNetCrossRefzbMATHADSGoogle Scholar
  20. 20.
    Nakatsu T., Takasaki K.: Melting crystal, quantum Torus and Toda hierarchy. Commun. Math. Phys. 285, 445–468 (2009)MathSciNetCrossRefzbMATHADSGoogle Scholar
  21. 21.
    Dickey L.A.: Lectures on classical W-algebras. Acta Appl. Math. 47, 243–321 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Witten E.: Two-dimensional gravity and intersection theory on moduli space. Surv. Differ. Geom. 1, 243–310 (1991)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kontsevich M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1–23 (1992)MathSciNetCrossRefzbMATHADSGoogle Scholar
  24. 24.
    Douglas M.: Strings in less than one dimension and the generalized KdV hierarchies. Phys. Lett. B 238, 176–180 (1990)MathSciNetCrossRefADSGoogle Scholar
  25. 25.
    Date, E., Kashiwara, M., Jimbo, M., Miwa, T.: Transformation groups for soliton equations. In: Nonlinear Integrable Systems—Classical Theory and Quantum Theory (Kyoto, 1981), 39–119. World Scientific Publishing, Singapore (1983)Google Scholar
  26. 26.
    Takasaki K.: Quasi-classical limit of BKP hierarchy and W-infinity symmetries. Lett. Math. Phys. 28, 177–185 (1993)MathSciNetCrossRefzbMATHADSGoogle Scholar
  27. 27.
    Tu M.H.: On the BKP hierarchy: additional symmetries, Fay identity and Adler-Shiota-van Moerbeke formula. Lett. Math. Phys. 81, 93–105 (2007)MathSciNetCrossRefzbMATHADSGoogle Scholar
  28. 28.
    Aoyama S., Kodama Y.: A generalized Sato equation and the W algebra. Phys. Lett. B 278, 56–62 (1992)MathSciNetCrossRefADSGoogle Scholar
  29. 29.
    Adler M., Shiota T., Moerbeke P.: From the w -algebra to its central extension: a τ-function approach. Phys. Lett. A 194, 33–43 (1994)MathSciNetCrossRefADSGoogle Scholar
  30. 30.
    Adler M., Shiota T., Moerbeke P.: A Lax representation for the vertex operator and the central extension. Commun. Math. Phys. 171, 547–588 (1995)CrossRefzbMATHADSGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsNingbo UniversityNingboPeople’s Republic of China

Personalised recommendations