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Sigma-Model Solitons on Noncommutative Spaces

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Abstract

We use results from time–frequency analysis and Gabor analysis to construct new classes of sigma-model solitons over the Moyal plane and over noncommutative tori, taken as source spaces, with a target space made of two points. A natural action functional leads to self-duality equations for projections in the source algebra. Solutions, having nontrivial topological content, are constructed via suitable Morita duality bimodules.

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References

  1. Connes A., Rieffel M.A.: Yang–Mills for non-commutative two-tori. Contemp. Math. 62, 237–266 (1987)

    Article  MathSciNet  Google Scholar 

  2. Dabrowski L., Krajewski T., Landi G.: Some properties of non-linear σ-models in noncommutative Geometry. Int. J. Mod. Phys. B 14, 2367–2382 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Dabrowski L., Krajewski T., Landi G.: Non-linear σ-models in noncommutative geometry: fields with values in finite spaces. Mod. Phys. Lett. A 18, 2371–2380 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  4. Daubechies I., Landau H.J., Landau Z.: Gabor time–frequency lattices and the Wexler–Raz identity. J. Fourier Anal. Appl. 1, 437–478 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. Farrell B., Strohmer T.: Inverse-closedness of a Banach algebra of integral operators on the Heisenberg group. J. Oper. Theory 64, 189–205 (2010)

    MATH  MathSciNet  Google Scholar 

  6. Feichtinger H.G., Kaiblinger N.: Varying the time–frequency lattice of Gabor frames. Trans. Am. Math. Soc. 356, 2001–2023 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Feichtinger, H.G., Strohmer, T.: Gabor Analysis and Algorithms. Theory and Applications. Birkhäuser, Boston (1998)

  8. Frank M., Larson D.R.: Frames in Hilbert C*-modules and C*-algebras. J. Oper. Theory 48, 273–314 (2002)

    MATH  MathSciNet  Google Scholar 

  9. Gabor D.: Theory of communication. J. IEE 93, 429–457 (1946)

    Google Scholar 

  10. Gayral V., Gracia-Bondia J.M., Iochum B., Schücker T., Varilly J.C.: Moyal planes are spectral triples. Commun. Math. Phys. 246, 569–623 (2004)

    Article  MATH  ADS  Google Scholar 

  11. Gröchenig K., Lyubarskii Y.: Gabor (super)frames with Hermite functions. Math. Ann. 345, 267–286 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gröchenig K., Stöckler J.: Gabor frames and totally positive functions. Duke Math. J. 162, 1003–1031 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Janssen A.J.E.M.: Duality and biorthogonality for Weyl–Heisenberg frames. J. Fourier Anal. Appl. 1, 403–436 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  14. Landi, G.: On harmonic maps in noncommutative geometry. In: Consani, C., Marcolli, M. (eds.) Noncommutative Geometry and Number Theory. Aspects of Mathematics E37, pp. 217–234. Vieweg, Wiesbaden (2006)

  15. Lee, H.H.: A note on non-linear σ-models in noncommutative geometry. arXiv:1410.5918

  16. Luef F.: Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces. J. Funct. Anal. 257, 1921–1946 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Luef F.: Projections in noncommutative tori and Gabor frames. Proc. Am. Math. Soc. 139, 571–582 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Luef F., Manin Y.I.: Quantum theta functions and Gabor frames for modulation spaces. Lett. Math. Phys. 88, 131–161 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Mathai V., Rosenberg J.: A noncommutative sigma-model. J. Noncommut. Geom. 5, 265–294 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rieffel M.A.: On the uniqueness of the Heisenberg commutation relations. Duke Math. J. 39, 745–752 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  21. Rieffel M.A.: Morita equivalence for C*-algebras and W*-algebras. J. Pure Appl. Algebra 5, 51–96 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rieffel M.A.: C*-algebras associated with irrational rotations. Pac. J. Math. 93, 415–429 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  23. Rieffel M.A.: Projective modules over higher-dimensional noncommutative tori. Can. J. Math. 40, 257–338 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  24. Rieffel M.A.: Vector bundles and Gromov–Hausdorff distance. J. K-Theory 5, 39–103 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. Schoenberg I.J.: On totally positive functions, Laplace integrals and entire functions of the Laguerre–Pólya-Schur type. Proc. Natl. Acad. Sci. USA 33, 11–17 (1947)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. Schoenberg I.J.: On Pólya frequency functions, I. The totally positive functions and their Laplace transforms. J. Anal. Math. 1, 331–374 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  27. Ron A., Shen Z.: Weyl–Heisenberg frames and Riesz bases in \({{L}_2(\mathbb{R}^d)}\). Duke Math. J. 89, 237–282 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  28. Toft, J.: Continuity and Schatten–von Neumann properties for pseudo-differential operators on modulation spaces. In: Toft, J., Wong, M., Zhu, H. (eds.) Modern Trends in Pseudo-Differential Operators. Operator Theory: Advances and Applications, vol. 172, pp. 173–206. Birkh äuser, Basel (2007)

  29. Wexler J., Raz S.: Discrete Gabor expansions. Signal Proc. 21, 207–220 (1990)

    Article  Google Scholar 

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Authors and Affiliations

  1. SISSA (Scuola Internazionale Superiore di Studi Avanzati), Via Bonomea 265, 34136, Trieste, Italy

    Ludwik Dabrowski

  2. Matematica, Università di Trieste, Via A. Valerio 12/1, 34127, Trieste, Italy

    Giovanni Landi

  3. INFN, Sezione di Trieste, Trieste, Italy

    Giovanni Landi

  4. Department of Mathematics, NTNU Trondheim, 7041, Trondheim, Norway

    Franz Luef

Authors
  1. Ludwik Dabrowski
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  2. Giovanni Landi
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  3. Franz Luef
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Correspondence to Giovanni Landi.

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Dabrowski, L., Landi, G. & Luef, F. Sigma-Model Solitons on Noncommutative Spaces. Lett Math Phys 105, 1663–1688 (2015). https://doi.org/10.1007/s11005-015-0790-x

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  • DOI: https://doi.org/10.1007/s11005-015-0790-x

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