Skip to main content

On harmonic maps in noncommutative geometry

  • Chapter
Noncommutative Geometry and Number Theory

Part of the book series: Aspects of Mathematics ((ASMA))

Abstract

We report on some recent work on harmonic maps and non-linear ζ-models in noncommutative geometry. After a general discussion we concentrate mainly on models on noncommutative tori.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A.A. Belavin, A.M. Polyakov, Metastable states of two-dimensional isotropic ferromagnets, JETP Lett. 22 (1975) 245–247.

    Google Scholar 

  2. F.-P. Boca, Projections in Rotation Algebras and Theta Functions, Commun. Math. Phys. 202 (1999) 325–357.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Connes, C*-algèbres et géométrie différentielle, C.R. Acad. Sci. Paris Sér. A 290 (1980) 599–604.

    MATH  MathSciNet  Google Scholar 

  4. A. Connes, Noncommutative Geometry, Academic Press, 1994.

    Google Scholar 

  5. A. Connes, Gravity coupled with matter and the foundation of noncommutative geometry, Commun. Math. Phys. 182 (1996) 155–176.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Connes, A short survey of noncommutative geometry, J.Math. Phys. 41 (2000) 3832–3866.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Connes, M. Douglas, A. Schwarz, Matrix theory compactification on tori, J.High Energy Phys. 02 (1998) 003.

    Article  MathSciNet  Google Scholar 

  8. A. Connes, M. Rieffel, Yang-Mills for Non-commutative Two-Tori, in Operator Algebras and Mathematical Physics, Contemp. Math. 62 (1987) 237–266.

    MathSciNet  Google Scholar 

  9. L. Dabrowski, T. Krajewski, G. Landi, Some Properties of Non-linear δ-models in Noncommutative Geometry; Int. J. Mod. Phys. B14 (2000) 2367–2382.

    MathSciNet  Google Scholar 

  10. L. Dabrowski, T. Krajewski, G. Landi, Non-linear δ-models in noncommutative geometry: fields with values in finite spaces, Mod. Phys. Lett. A18 (2003) 2371–2380.

    MathSciNet  Google Scholar 

  11. M. Dieng, A. Schwarz, Differential and complex geometry of two-dimensional noncommutative tori, Lett. Math. Phys. 61 (2002) 263–270.

    Article  MATH  MathSciNet  Google Scholar 

  12. T. Krajewski, Gauge invariance of the Chern-Simons action in noncommutative geometry, ISI GUCCIA Conference ‘Quantum Groups, Noncommutative Geometry and Fundamental Physical Interactions’, Palermo December 1997, math-ph/9810015.

    Google Scholar 

  13. J. Eells, Harmonic maps: Selected papers of James Eells and collaborators, World Scientific, 1992.

    Google Scholar 

  14. K. Gawedzki, Lectures on conformal field theory, in ‘Quantum Fields and Strings: a Course for Mathematicians’, P. Deligne et al. editors, American Mathematical Society 1999; pp 727–805.

    Google Scholar 

  15. M. Green, J.H. Schwarz, E. Witten, Superstring theory, Cambridge University Press, 1987.

    Google Scholar 

  16. Yu. I. Manin, Quantized theta-function, Prog. Theor. Phys. Suppl. 102 (1990) 219–228.

    Article  MathSciNet  Google Scholar 

  17. Yu. I. Manin, Theta functions, quantum tori and Heisenberg group, Lett. Math. Phys. 56 (2001) 295–320.

    Article  MATH  MathSciNet  Google Scholar 

  18. Yu. I. Manin, Real multiplication and noncommutative geometry, math.QA/0202109.

    Google Scholar 

  19. Yu. I. Manin. Functional equations for quantum theta functions, math.QA/0307393.

    Google Scholar 

  20. M. Pimsner, D. Voiculescu, Exact Sequences for K-Groups and Ext-Groups of Certain Cross-Product C*-Algebras, J. Oper. Theory 4 (1980) 93–118.

    MATH  MathSciNet  Google Scholar 

  21. J. Polchinski, String theory, Cambridge University Press, 1998.

    Google Scholar 

  22. A. Polishchuk, Analogues of the exponential map associated with complex structures on noncommutative two-tori, math.QA/0404056.

    Google Scholar 

  23. A. Polishchuk, A. Schwarz, Categories of holomorphic vector bundles on noncommutative two-tori, Commun. Math. Phys. 236 (2003) 135–159.

    Article  MATH  MathSciNet  Google Scholar 

  24. M. Rieffel, C*-algebras associated with irrational rotations, Pacific J. Math. 93 (1981) 415–429.

    MATH  MathSciNet  Google Scholar 

  25. M. Rieffel, The Cancellation Theorem for projective Modules over irrational C*-algebras, Proc. London Math. Soc. 47 (1983) 1285–302.

    Article  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  27. M. Rieffel, Non-commutative Tori-A case study of Non-commutative Differentiable Manifolds Contemp. Math. 105 (1991) 191–211.

    MathSciNet  Google Scholar 

  28. N. Seiberg, E. Witten, String Theory and Noncommutative Geometry, JHEP 09 (1999) 032.

    Article  MathSciNet  Google Scholar 

  29. M. Spera, Yang-Mills theory in non commutative differential geometry, Rend. Sem. Fac. Scienze Univ. Cagliari, Suppl. 58 (1988) 409–421.

    MathSciNet  Google Scholar 

  30. A. Schwarz, Theta-functions on noncommutative tori, Lett. Math. Phys. 58 (2001) 81–90.

    Article  MATH  MathSciNet  Google Scholar 

  31. J.C. Varilly, An Introduction to Noncommutative Geometry, Lectures at EMS Summer School on NCG and Applications, Sept 1997, physics/9709045.

    Google Scholar 

  32. S. Walters, The AF Structure of Noncommutative Toroidal ℤ/4ℤ Orbifolds, J. Reine Angew. Math. 568 (2004) 139–196.

    MATH  MathSciNet  Google Scholar 

  33. W.J. Zakrzewski, Low dimensional sigma models, Adam Hilger, Bristol 1989.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Additional information

This paper is dedicated to Marta.

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Friedr. Vieweg & Sohn Verlag ∣ GWV Fachverlage GmbH, Wiesbaden

About this chapter

Cite this chapter

Landi, G. (2006). On harmonic maps in noncommutative geometry. In: Consani, C., Marcolli, M. (eds) Noncommutative Geometry and Number Theory. Aspects of Mathematics. Vieweg. https://doi.org/10.1007/978-3-8348-0352-8_11

Download citation

Publish with us

Policies and ethics