Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 058, 28 pages      arXiv:1003.5590      https://doi.org/10.3842/SIGMA.2010.058
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors

Horatiu Nastase a and Constantinos Papageorgakis b
a) Instituto de Física Teórica, UNESP-Universidade Estadual Paulista, R. Dr. Bento T. Ferraz 271, Bl. II, Sao Paulo 01140-070, SP, Brazil
b) Department of Mathematics, King's College London, The Strand, London WC2R 2LS, UK

Received March 26, 2010, in final form July 09, 2010; Published online July 20, 2010

Abstract
We review our construction of a bifundamental version of the fuzzy 2-sphere and its relation to fuzzy Killing spinors, first obtained in the context of the ABJM membrane model. This is shown to be completely equivalent to the usual (adjoint) fuzzy sphere. We discuss the mathematical details of the bifundamental fuzzy sphere and its field theory expansion in a model-independent way. We also examine how this new formulation affects the twisting of the fields, when comparing the field theory on the fuzzy sphere background with the compactification of the 'deconstructed' (higher dimensional) field theory.

Key words: noncommutative geometry; fuzzy sphere; field theory.

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References

  1. Myers R.C., Dielectric-branes, J. High Energy Phys. 1999 (1999), no. 12, 022, 41 pages, hep-th/9910053.
  2. Hoppe J., Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, PhD Thesis, Massachusetts Institute of Technology, 1982, available at http://hdl.handle.net/1721.1/15717.
  3. Hoppe J., Diffeomorphism groups, quantization and SU(∞), Internat. J. Modern Phys. A 4 (1989), 5235-5248.
  4. Madore J., The fuzzy sphere, Classial Quantum Gravity 9 (1992), 69-88.
  5. Iso S., Kimura Y., Tanaka K., Wakatsuki K., Noncommutative gauge theory on fuzzy sphere from matrix model, Nuclear Phys. B 604 (2001), 121-147, hep-th/0101102.
  6. Dasgupta K., Sheikh-Jabbari M.M., Van Raamsdonk M., Matrix perturbation theory for M-theory on a PP-wave, J. High Energy Phys. 2002 (2002), no. 5, 056, 52 pages, hep-th/0205185.
  7. Dasgupta K., Sheikh-Jabbari M.M., Van Raamsdonk M., Protected multiplets of M-theory on a plane wave, J. High Energy Phys. 2002 (2002), no. 9, 021, 41 pages, hep-th/0207050.
  8. Papageorgakis C., Ramgoolam S., Toumbas N., Noncommutative geometry, quantum effects and DBI-scaling in the collapse of D0-D2 bound states, J. High Energy Phys. 2006 (2006), no. 1, 030, 31 pages, hep-th/0510144.
  9. Nastase H., Papageorgakis C., Ramgoolam S., The fuzzy S2 structure of M2-M5 systems in ABJM membrane theories, J. High Energy Phys. 2009 (2009), no. 5, 123, 61 pages, arXiv:0903.3966.
  10. Nastase H., Papageorgakis C., Fuzzy Killing spinors and supersymmetric D4 action on the fuzzy 2-sphere from the ABJM model, J. High Energy Phys. 2009 (2009), no. 12, 049, 52 pages, arXiv:0908.3263.
  11. Aharony O., Bergman O., Jafferis D.L., Maldacena J., N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, J. High Energy Phys. 2008 (2008), no. 10, 091, 38 pages, arXiv:0806.1218.
  12. Bagger J., Lambert N., Modeling multiple M2-branes, Phys. Rev. D 75 (2007), 045020, 7 pages, hep-th/0611108.
  13. Bagger J., Lambert N., Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D 77 (2008), 065008, 6 pages, arXiv:0711.0955.
  14. Bagger J., Lambert N., Comments on multiple M2-branes, J. High Energy Phys. 2008 (2008), no. 2, 105, 15 pages, arXiv:0712.3738.
  15. Gustavsson A., Algebraic structures on parallel M2-branes, Nuclear Phys. B 811 (2009), 66-76, arXiv:0709.1260.
  16. Gustavsson A., Rey S.-J., Enhanced N=8 supersymmetry of ABJM theory on R(8) and R(8)/Z(2), arXiv:0906.3568.
  17. Kwon O.-K., Oh P., Sohn J., Notes on Supersymmetry Enhancement of ABJM theory, J. High Energy Phys. 2009 (2009), no. 8, 093, 22 pages, arXiv:0906.4333.
  18. Gomis J., Rodríguez-Gómez D., Van Raamsdonk M., Verlinde H., A massive study of M2-brane proposals, J. High Energy Phys. 2008 (2008), no. 9, 113, 29 pages, arXiv:0807.1074.
  19. Hosomichi K., Lee K.-M., Lee S., Lee S., Park J., N=5,6 superconformal Chern-Simons theories and M2-branes on orbifolds, J. High Energy Phys. 2008 (2008), no. 9, 002, 24 pages, arXiv:0806.4977.
  20. Bena I., Warner N.P., A harmonic family of dielectric flow solutions with maximal supersymmetry, J. High Energy Phys. 2004 (2004), no. 12, 021, 22 pages, hep-th/0406145.
  21. Lin H., Lunin O., Maldacena J.M., Bubbling AdS space and 1/2 BPS geometries, J. High Energy Phys. 2004 (2004), no. 10, 025, 68 pages, hep-th/0409174.
  22. Terashima S., On M5-branes in N=6 membrane action, J. High Energy Phys. 2008 (2008), no. 8, 080, 11 pages, arXiv:0807.0197.
  23. Hanaki K., Lin H., M2-M5 systems in N=6 Chern-Simons theory, J. High Energy Phys. 2008 (2008), no. 9, 067, 14 pages, arXiv:0807.2074.
  24. Guralnik Z., Ramgoolam S., On the polarization of unstable D0-branes into non-commutative odd spheres, J. High Energy Phys. 2001 (2001), no. 2, 032, 17 pages, hep-th/0101001.
  25. Ramgoolam S., On spherical harmonics for fuzzy spheres in diverse dimensions, Nuclear Phys. B 610 (2001), 461-488, hep-th/0105006.
  26. Ramgoolam S., Higher dimensional geometries related to fuzzy odd-dimensional spheres, J. High Energy Phys. 2002 (2002), no. 10, 064, 29 pages, hep-th/0207111.
  27. Van Raamsdonk M., Comments on the Bagger-Lambert theory and multiple M2-branes, J. High Energy Phys. 2008 (2008), no. 5, 105, 9 pages, arXiv:0803.3803.
  28. Hasebe K., Kimura Y., Fuzzy supersphere and supermonopole, Nuclear Phys. B 709 (2005), 94-114, hep-th/0409230.
  29. Grosse H., Reiter G., The fuzzy supersphere, J. Geom. Phys. 28 (1998), 349-383, math-ph/9804013.
  30. Balachandran A.P., Kurkcuoglu S., Vaidya S., Lectures on fuzzy and fuzzy SUSY physics, hep-th/0511114.
  31. van Nieuwenhuizen P., An introduction to simple supergravity and the Kaluza-Klein program, in Relativity, Groups and Topology, II (Les Houches, 1983), North-Holland, Amsterdam, 1984, 823-932.
  32. Eastaugh A., van Nieuwenhuizen P., Harmonics and spectra on general coset manifolds, Kyoto Summer Institute 1985:0001, Preprint ITP-SB-85-43.
  33. van Nieuwenhuizen P., The complete mass spectrum of d=11 supergravity compactified on S4 and a general mass formula for arbitrary cosets M4, Classial Quantum Gravity 2 (1985), 1-20.
  34. Gunaydin M., van Nieuwenhuizen P., Warner N.P., General construction of the unitary representations of anti-de Sitter superalgebras and the spectrum of the S4 compactification of 11-dimensional supergravity, Nuclear Phys. B 255 (1985), 63-92.
  35. Nastase H., Vaman D., van Nieuwenhuizen P., Consistency of the AdS7×S4 reduction and the origin of self-duality in odd dimensions, Nuclear Phys. B 581 (2000), 179-239, hep-th/9911238.
  36. Van Nieuwenhuizen P., Supergravity, Phys. Rep. 68 (1981), 189-398.
  37. Kim H.J., Romans L.J. , van Nieuwenhuizen P., Mass spectrum of chiral ten-dimensional N=2 supergravity on S5, Phys. Rev. D 32 (1985), 389-399.
  38. Wu Y.S., Zee A., Membranes, higher Hopf maps, and phase interactions, Phys. Lett. B 207 (1988), 39-43.
  39. Bernevig B.A., Hu J.-P., Toumbas N., Zhang S.-C., Eight-dimensional quantum hall effect and "octonions", Phys. Rev. Lett. 91 (2003), 236803, 4 pages, cond-mat/0306045.
  40. Arkani-Hamed N., Cohen A.G., Georgi H., (De)constructing dimensions, Phys. Rev. Lett. 86 (2001), 4757-4761, hep-th/0104005.
  41. Andrews R.P., Dorey N., Deconstruction of the Maldacena-Núñez compactification, Nuclear Phys. B 751 (2006), 304-341, hep-th/0601098.
  42. Maldacena J.M., Nuñez C., Supergravity description of field theories on curved manifolds and a no go theorem, Internat. J. Modern Phys. A 16 (2001), 822-855, hep-th/0007018.
  43. Bershadsky M., Vafa C., Sadov V., D-branes and topological field theories, Nuclear Phys. B 463 (1996), 420-434, hep-th/9511222.
  44. Mukhi S., Papageorgakis C., M2 to D2, J. High Energy Phys. 2008 (2008), no. 5, 085, 15 pages, arXiv:0803.3218.
  45. Maldacena J., Martelli D., The unwarped, resolved, deformed conifold: fivebranes and the baryonic branch of the Klebanov-Strassler theory, arXiv:0906.0591.

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