Letters in Mathematical Physics

, Volume 54, Issue 2, pp 123–135 | Cite as

Moyal Deformation, Seiberg–Witten Maps, and Integrable Models

  • A. Dimakis
  • F. Müller-Hoissen


A covariant formalism for Moyal deformations of gauge theory and differential equations which determine Seiberg–Witten maps is presented. Replacing the ordinary product of functions by the noncommutative Moyal product, noncommutative versions of integrable models can be constructed. We explore how a Seiberg–Witten map acts in such a framework. As a specific example, we consider a noncommutative extension of the principal chiral model.

deformation quantization star product integrable model Seiberg–Witten map 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Seiberg, N. and Witten, E.: String theory and noncommutative geometry, J. High Energy Phys. 9 (1999), 32.Google Scholar
  2. 2.
    Asakawa, T. and Kishimoto, I.: Comments on gauge equivalence in noncommutative geometry, J. High Energy Phys. 11 (1999), 24.Google Scholar
  3. 3.
    Cornalba, L.: D-brane physics and noncommutative Yang-Mills theory, hep-th/ 9909081; Ishibashi, N.: A relation between commutative and noncommutative descriptions of D-branes, hep-th/9909176; Ishibashi, N., Iso, S., Kawai, H. and Kitazawa, Y.: Wilson loops in non-commutative Yang-Mills, Nuclear Phys. B 573 (2000), 573; Okuyama, K.: A path integral representation of the map between commutative and noncommutative gauge fields, J. High Energy Phys. 3 (2000), 16; Chu, C.-S., Ho, P.-M. and Li, M.: Matrix theory in a constant C field background, Nuclear Phys. B 574 (2000), 275; Andreev, O. and Dorn, H.: On open string sigma-model and noncommutative gauge fields, hep-th/9912070; Jurco, B. and Schupp, P.: Noncommutative Yang-Mills from equivalence of star products, hep-th/0001032; Terashima, S.: On the equivalence between noncommutative and ordinary gauge theories, hep-th/0001111, to appear in J. High Energy Phys.; Madore, J., Schraml, S., Schupp, P. and Wess, J.: Gauge theory on noncommutative spaces, hep-th/0001203; Hashimoto, K. and Hirayama, T.: Branes and BPS configurations of non-commutative/commutative gauge theories, hep-th/0002090; Asakawa, T. and Kishimoto, I.: Noncommutative gauge theories from deformation quantization, hep-th/0002138; Alekseev, A. Y. and Bytsko, A. G.: Wilson lines on noncommutative tori, Phys. Lett. B 482 (2000), 271; Moriyama, S.: Noncommutative monopole from nonlinear monopole, hep-th/0003231; Benaoum, H. B.: On noncommutative and commutative equivalence for BFYM theory: Seiberg-Witten map, hep-th/0004002; Jurco, B., Schupp, P. and Wess, J.: Noncommutative gauge theory for Poisson manifolds, hep-th/0005005; Goto, S. and Hata, H.: Noncommutative monopole at the second order in θ, hep-th/0005101; Terashima, S.: The non-Abelian Born-Infeld action and noncommutative gauge theory, hep-th/0006058; Jurco, B., Schraml, S., Schupp, P. and Wess, J.: Enveloping algebra valued gauge transformations for nonabelian gauge groups on noncommutative spaces, hep-th/0006246; Rey, S.-J. and von Unge, R.: S-duality, noncritical open string and noncommutative gauge theory, hep-th/0007089.Google Scholar
  4. 4.
    Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A. and Sternheimer, D.: Deformation theory and quantization I, II, Ann. Phys. 111 (1978), 61.Google Scholar
  5. 5.
    Dimakis, A. and Müller-Hoissen, F.: Bi-differential calculi and integrable models, J. Phys. A 33 (2000), 957; Bicomplexes and integrable models, J. Phys. A 33 (2000), 6579.Google Scholar
  6. 6.
    Dimakis, A. and Müller-Hoissen, F.: Bicomplexes, integrable models, and noncommutative geometry, hep-th/0006005; A noncommutative version of the nonlinear Schrüdinger equation, hep-th/0007015.Google Scholar
  7. 7.
    Dimakis, A. and Müller-Hoissen, F.: The Korteweg-de-Vries equation on a noncommutative space-time, Phys. Lett. A 278 (2001) 139.Google Scholar
  8. 8.
    Takasaki, K.: Anti-self dual Yang-Mills equations on noncommutative spacetime, hep-th/0005194.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • A. Dimakis
    • 1
  • F. Müller-Hoissen
    • 2
  1. 1.Department of MathematicsUniversity of the AegeanKarlovasi, SamosGreece
  2. 2.Max-Planck-Institut für StrömungsforschungGöttingenGermany

Personalised recommendations