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Journal of Mathematical Physics / Volume 53 / Issue 2 / ARTICLES / Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

J. Math. Phys. 53, 022303 (2012); http://dx.doi.org/10.1063/1.3679398 (24 pages)

Noncommutative deformation of spinor zero mode and Atiyah-Drinfeld-Hitchin-Manin construction

Yoshiaki Maeda1 and Akifumi Sako2

1Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan and Mathematical Research Centre, University of Warwick, Coventry, CV4 7AL , United Kingdom
2Department of General Education, Kushiro National College of Technology, Otanoshike-Nishi 2-32-1, Kushiro 084-0916, Japan

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(Received 1 April 2011; accepted 4 January 2012; published online 10 February 2012)

A method to construct noncommutative instantons as deformations from commutative instantons was provided by Maeda and Sako [J. Geom. Phys. 58, 1784 (2008)]10.1016/j.geomphys.2008.08.006. Using this noncommutative deformed instanton, we investigate the spinor zero modes of the Dirac operator in a noncommutative instanton background on noncommutative math4, and we modify the index of the Dirac operator on the noncommutative space slightly and show that the number of the zero mode of the Dirac operator is preserved under the noncommutative deformation. We prove the existence of the Green's function associated with instantons on noncommutative math4, as a smooth deformation of the commutative case. The feature of the zero modes of the Dirac operator and the Green's function derives noncommutative ADHM (Atiyah-Drinfeld-Hitchin-Manin) equations which coincide with the ones introduced by Nekrasov and Schwarz. We show a one-to-one correspondence between the instantons on noncommutative math4 and ADHM data. An example of a noncommutative instanton and a spinor zero mode are also given.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. NOTATIONS, DEFINITIONS, AND KNOWN FACTS
  3. THE INDEX OF THE DIRAC OPERATOR
  4. GREEN'S FUNCTION
  5. FROM INSTANTONS TO THE ADHM EQUATIONS
  6. EXAMPLE
  7. CONCLUSION

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KEYWORDS and PACS

Keywords

Dirac equation, Green's function methods, noncommutative field theory

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3679398

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    N. H. Christ, E. J. Weinberg, and N. K. Stanton, Phys. Rev. D 18, 2013 (1978).

    L. S. Brown, R. D. Carlitz, D. B. Creamer, and C. k. Lee, Phys. Rev. D 17, 1583 (1978).


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