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Space-Time Foam from Non-Commutative Instantons

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Abstract

We show that a U(1) instanton on non-commutative corresponds to a non-singular U(1) gauge field on a commutative Kähler manifold X which is a blowup of at a finite number of points. This gauge field on X obeys Maxwell’s equations in addition to the susy constraint F0,2=0. For instanton charge k the manifold X can be viewed as a space-time foam with b2∼k. A direct connection with integrable systems of Calogero-Moser type is established. We also make some comments on the non-abelian case.

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References

  1. Nekrasov, N., Schwarz, A.S.: Instantons on Noncommutative. Commun. Math. Phys. 198, 689 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Nekrasov, N.: Topological theories and Zonal Spherical Functions. ITEP publications, 1995 (in Russian)

  3. Wilson, G.: Collisions of Calogero-Moser particles and adelic Grassmannian. Invent. Math. 133, 1–41 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. See the contributions of Wilson, G., Krichever, I., Nekrasov, N., Braden, H.W. In: Proceedings of the Workshop on Calogero-Moser-Sutherland models CRM Series in Mathematical Physics, New York: Springer-Verlag, 2000

  5. Gorsky, A., Krichever, I., Marshakov, A., Mironov, A., Morozov, A.: Integrability and Seiberg-Witten Exact Solution. Phys.Lett. B355, 466–474 (1995)

    Google Scholar 

  6. Martinec, E., Warner, N.: Integrable systems and supersymmetric gauge theory. Nucl.Phys. 459, 97–112 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Donagi, R., Witten, E.: Supersymmetric Yang-Mills Systems And Integrable Systems. Nucl.Phys. B460, 299–344 (1996)

    Google Scholar 

  8. Gorsky, A., Gukov, S., Mironov, A.: Multiscale N=2 SUSY field theories, integrable systems and their stringy/brane origin – I. Nucl.Phys. B517, 409–461 (1998)

    Google Scholar 

  9. Braden, H.W., Marshakov, A., Mironov, A., Morozov, A.: The Ruijsenaars-Schneider Model in the Context of Seiberg-Witten Theory. Nucl. Phys. B558, 371–390 (1999)

    Google Scholar 

  10. See for example contributions in, Integrability: the Seiberg-Witten and Whitham equations, eds H.W. Braden and I.M. Krichever, Amsterdam: Gordon and Breach Science Publishers, 2000

  11. Gorsky, A., Nekrasov, N., Rubtsov, V.: Hilbert Schemes, Separated Variables, and D-branes. Commun. Math. Phys. 222, 299–318 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Aharony, O., Berkooz, M., Kachru, S., Seiberg, N., Silverstein, E.: Matrix Description of Interacting Theories in Six Dimensions. Adv.Theor.Math.Phys. 1, 148–157 (1998)

    MATH  Google Scholar 

  13. Aharony, O., Berkooz, M., Seiberg, N.: Light-Come Description of(Z.O) Superconformal Theories in six Dimensions. Adv.Theor.Math.Phys. 2, 119–153 (1998)

    MATH  Google Scholar 

  14. Witten, E., Seiberg, N.: String Theory and Noncommutative Geometry. JHEP 9909, 032 (1999)

    Google Scholar 

  15. Douglas, M.: Branes within Branes. http://arxiv.org.labs/hep-th/9512077, 1995

  16. Douglas, M.: Gauge Fields and D-branes. J. Geom. Phys. 28, 255–262 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Corrigan, E., Goddard, P.: Construction of instanton and monopole solutions and reciprocity. 154, 253 (1984)

  18. Hitchin, N.J., Karlhede, A., Lindstrom, U., Rocek, M.: Hyperkähler Metrics and Supersymmetry. Commun. Math. Phys. 108, 535 (1987)

    MathSciNet  MATH  Google Scholar 

  19. Losev, A., Nekrasov, N., Shatashvili, S.: The Freckled Instantons. In: The many Faces of The Superworld, Y. Golfand Memorial Volume, M. Shifman ed., Singapore: World Scientific, 2000

  20. Nakajima, H.: Lectures on Hilbert Schemes of Points on Surfaces. AMS University Lecture Series, Providence, RI: AMS, 1999

  21. Kazhdan, D., Kostant, B., Sternberg, S.: Hamiltonian Group Actions and Dynamical Systems of Calogero Type. Commun. Pure and Appl. Math. 31, 481–507 (1978)

    MATH  Google Scholar 

  22. Nekrasov, N.: On a duality in Calogero-Moser-Sutherland systems. http://arxiv.org/abs/hep-th/9707111, 1997

  23. Fock, V., Gorsky, A., Nekrasov, N., Rubtsov, V.: Duality in Integrable Systems and Gauge Theories. JHEP 0007, 028 (2000)

    Google Scholar 

  24. Ruijsenaars, S. M.: Complete Integrability of Relativistic Calogero-Moser Systems and Elliptic Function Indentities. Comm. Math. Phys. 110, 191–213 (1987); S. M. Ruijsenaars, H. Schneider: Ann. Phys. (NY) 170, 370 (1986)

    MathSciNet  MATH  Google Scholar 

  25. Gorsky, A., Nekrasov, N.: Relativistic Calogero-Moser Modelas gauged WZW Theory. Nucl. Phys. B436, 582 (1995)

  26. Terashima, S.: Instantons in the U(1) Born-Infeld Theory and Noncommutative Gauge Theory. Phys. Lett. 477B, 292–298 (2000); M. Mariño, R. Minasian, G. Moore, A. Strominger: Non-linear Instantons from Supersymmetric p-Branes. JHEP 0001, 005 (2000)

    MATH  Google Scholar 

  27. Furuuchi, K.: Instantons on Noncommutative R4 and Projection Operators. Prog. Theor. Phys. 103, 1043–1068 (2000); Equivalence of projections as Gauge Equivalence on Noncommutative Space. Commun. Math. Phys. 217, 579–593 (2001); Topological charge of U(I) Instantons. Prog. Theor. Phys. Suppl. 144, 79–91 (2001)

    MathSciNet  Google Scholar 

  28. Nekrasov, N.: Noncommutative instantons revisited. Commun. Math. Phys. 241, 143–160 (2003)

    MATH  Google Scholar 

  29. Burns, D.: In: Twistors and Harmonic Maps. Lecture. Amer. Math. Soc. Conference, Charlotte, NC, 1986

  30. Iqbal, A., Nekrasov, N., Okounkov, A., Vafa, C.: Quantum foam and topological strings. http:// arxiv.org/abs/hep-th/0312022, 2003

  31. Ishikawa, T., Kuroki, S.-I., Sako, A.: Elongated U(I) Instantons on Noncommutative Rr. JHEP 111, 068 (2001)

    Google Scholar 

  32. Kraus, P., Shigemori, M.: Non-Commutative Instantons and the Seiberg-Witten Map. JHEP 0206, 034 (2002)

    Article  Google Scholar 

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Authors and Affiliations

  1. Dept. of Mathematics and Statistics, University of Edinburgh, Edinburgh, EH9 3JZ, Scotland

    Harry W Braden

  2. Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA

    Nikita A. Nekrasov

  3. Institute for Theoretical and Experimental Physics, 117259, Moscow, Russia

    Harry W Braden & Nikita A. Nekrasov

  4. IHES, 91440, Bures-sur-Yvette, France

    Nikita A. Nekrasov

Authors
  1. Harry W Braden
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  2. Nikita A. Nekrasov
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Correspondence to Harry W Braden.

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Communicated by M.R. Douglas

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Braden, H., Nekrasov, N. Space-Time Foam from Non-Commutative Instantons. Commun. Math. Phys. 249, 431–448 (2004). https://doi.org/10.1007/s00220-004-1127-2

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  • DOI: https://doi.org/10.1007/s00220-004-1127-2

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