Commutativity equations and dressing transformations

  • A. Losev
  • I. Polyubin
Fields, Particles, and Nuclei


We study dressing transformations that generate all solutions to commutativity equations and, after picking up special coordinates, all solutions to WDVV equations. We conjecture that the homological tensor product of solutions to the commutativity equations corresponds to the tensor product of matrices of the dressing transformation and check this in the first nontrivial case.

PACS numbers



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Witten, Nucl. Phys. B 340, 281 (1990).CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    R. Dijkgraaf and E. Witten, Nucl. Phys. B 342, 486 (1990).CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    E. Verlinde and H. Verlinde, Nucl. Phys. B 348, 457 (1991).CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    R. Dijkgraaf, E. Verlinde, and H. Verlinde, in Proceedings of the Trieste Spring School on String Theory and Quantum Gravity, 1990, Ed. by M. Green et al. (World Sci., Singapore, 1991).Google Scholar
  5. 5.
    E. Witten, hep-th/9207094.Google Scholar
  6. 6.
    A. Losev, Theor. Math. Phys. 95, 595 (1993).CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    T. Eguchi, H. Kanno, Y. Yamada, et al., Phys. Lett. B 305, 235 (1993).ADSMathSciNetGoogle Scholar
  8. 8.
    M. Kontsevich and Yu. Manin, Commun. Math. Phys. 164, 525 (1994).CrossRefMathSciNetGoogle Scholar
  9. 9.
    B. Dubrovin, hep-th/9407018; Lect. Notes Math. 1620, 120 (1996).zbMATHMathSciNetGoogle Scholar
  10. 10.
    M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Commun. Math. Phys. 165, 311 (1994).CrossRefMathSciNetGoogle Scholar
  11. 11.
    A. Losev and I. Polyubin, Int. J. Mod. Phys. A 10, 4161 (1995).ADSMathSciNetGoogle Scholar
  12. 12.
    M. Kontsevich and Yu. Manin, with appendix by Kaufmann, q-alg/9502009.Google Scholar
  13. 13.
    N. Berkovits, J. High Energy Phys. 0004, 018 (2000).Google Scholar
  14. 14.
    S. Cecotti and C. Vafa, Nucl. Phys. B 367, 359 (1991).CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    A. Losev, in Proceedings of Taniguchi Conference on Theory of Primitive Form and Topological Field Theory; hep-th/9801179.Google Scholar
  16. 16.
    A. Losev, Preprint No. ITEP-TH-84/98, ITÉF (Inst. of Theoretical and Experimental Physics, Moscow, 1998); Preprint LPTHE-61/98 (1998).Google Scholar
  17. 17.
    A. Losev and Yu. Manin, Mich. Math. J. 48, 443 (2000).MathSciNetGoogle Scholar
  18. 18.
    A. Losev and I. Polyubin, Pis’ma Zh. Éksp. Teor. Fiz. 73, 59 (2001) [JETP Lett. 73, 53 (2001)].Google Scholar
  19. 19.
    A. Givental,, AG/0008067.Google Scholar
  20. 20.
    B. Dubrovin and Y. Zhang,, DG/0108160.Google Scholar
  21. 21.
    H. Aratyn and J. Van de Leur, in Talk on 15th Euroconference on Nonlinear Evolution Equations and Dynamical Systems NEEDS 2001, Cambridge, England, 2001; hep-th/0111243.Google Scholar

Copyright information

© MAIK "Nauka/Interperiodica" 2003

Authors and Affiliations

  • A. Losev
    • 1
  • I. Polyubin
    • 1
    • 2
  1. 1.Institute of Theoretical and Experimental PhysicsMoscowRussia
  2. 2.Landau Institute for Theoretical PhysicsMoscowRussia

Personalised recommendations