Theoretical and Mathematical Physics

, Volume 153, Issue 3, pp 1643–1651 | Cite as

Noncommutative Grassmannian U(1) sigma model and a Bargmann-Fock space

  • A. V. Komlov


We consider a Grassmannian version of the noncommutative U(1) sigma model specified by the energy functional E(P) = ‖[a, P]‖ HS 2 , where P is an orthogonal projection operator in a Hilbert space H and a: H → H is the standard annihilation operator. With H realized as a Bargmann-Fock space, we describe all solutions with a one-dimensional range and prove that the operator [a, P] is densely defined in H for a certain class of projection operators P with infinite-dimensional ranges and kernels.


noncommutative U(1) sigma model Bargmann-Fock space 


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  1. 1.
    W. J. Zakrzewski, Low Dimensional Sigma Models, Adam Hilger, Bristol (1989).MATHGoogle Scholar
  2. 2.
    J. A. Harvey, “Komaba lectures on noncommutative solitons and D-branes,” arXiv:hep-th/0102076v1 (2001).Google Scholar
  3. 3.
    O. Lechtenfeld and A. D. Popov, JHEP, 0111, 040 (2001); arXiv:hep-th/0106213v3 (2001).CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    A. V. Domrin, O. Lechtenfeld, and S. Petersen, JHEP, 0503, 045 (2005); arXiv: hep-th/0412001v2 (2004).CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    L. Hörmander, The Analysis of Linear Partial Differential Operators: III. Pseudodifferential Operators (Grundlehren Math. Wiss., Vol. 274), Springer, Berlin (1985).Google Scholar
  6. 6.
    V. Bargmann, Comm. Pure Appl. Math., 14, 187 (1961).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    F. E. Burstall and J. H. Rawnsley, Comm. Math. Phys., 110, 311 (1987).MATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    D. J. Newman and H. S. Shapiro, “Fisher spaces of entire functions,” in: Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., Vol. 11), Amer. Math. Soc., Providence, R. I. (1968), p. 360.Google Scholar
  9. 9.
    N. I. Ahiezer, Lectures in the Theory of Approximation [in Russian], Nauka, Moscow (1965).Google Scholar

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© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

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