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

SIGMA 2 (2006), 081, 14 pages      hep-th/0609205
Contribution to the Proceedings of the O'Raifeartaigh Symposium

Quantum Field Theory in a Non-Commutative Space: Theoretical Predictions and Numerical Results on the Fuzzy Sphere

Marco Panero
School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland

Received September 29, 2006, in final form November 10, 2006; Published online November 17, 2006

We review some recent progress in quantum field theory in non-commutative space, focusing onto the fuzzy sphere as a non-perturbative regularisation scheme. We first introduce the basic formalism, and discuss the limits corresponding to different commutative or non-commutative spaces. We present some of the theories which have been investigated in this framework, with a particular attention to the scalar model. Then we comment on the results recently obtained from Monte Carlo simulations, and show a preview of new numerical data, which are consistent with the expected transition between two phases characterised by the topology of the support of a matrix eigenvalue distribution.

Key words: non-commutative geometry; matrix models; non-perturbative effects; phase transitions.

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  1. Doplicher S., Fredenhagen K., Roberts J.E., The quantum structure of space-time at the Planck scale and quantum fields, Comm. Math. Phys., 1995, V.172, 187-220, hep-th/0303037.
  2. Landi G., An introduction to noncommutative spaces and their geometry, hep-th/9701078.
  3. Douglas M.R., Nekrasov N.A., Noncommutative field theory, Rev. Modern Phys., 2001, V.73, 977-1029, hep-th/0106048.
  4. Szabo R.J., Symmetry, gravity and noncommutativity, hep-th/0606233.
  5. Karabali D., Nair V.P., Randjbar-Daemi S., Fuzzy spaces, the M(atrix) model and the quantum Hall effect, hep-th/0407007.
  6. Connes A., Douglas M.R., Schwarz A.S., Noncommutative geometry and matrix theory: Compactification on tori, JHEP, 1998, 9802, 003, 35 pages, hep-th/9711162.
  7. Douglas M.R., Hull C.M., D-branes and the noncommutative torus, JHEP, 1998, N 2, Paper 008, 5 pages, hep-th/9711165.
  8. Alekseev A.Y., Recknagel A., Schomerus V., Non-commutative world-volume geometries: Branes on SU(2) and fuzzy spheres, JHEP, 1999, N 9, Paper 023, 20 pages, hep-th/9908040.
  9. Seiberg N., Witten E., String theory and noncommutative geometry, JHEP, 1999, N 9, Paper 032, 93 pages, hep-th/9908142.
  10. Kar S., Non-commutativity, zero modes and D-brane geometry, Nuclear Phys. B, 2000, V.577, 171-182, hep-th/9911251.
  11. Kar S., D-branes, cyclic symmetry and noncommutative geometry, Modern Phys. Lett. A, 2003, V.18, 1053-1065, hep-th/0006073.
  12. Chamseddine A.H., Felder G., Fröhlich J., Gravity in noncommutative geometry, Comm. Math. Phys., 1993, V.155, 205-217, hep-th/9209044.
  13. Groenewold H.J., On the principles of elementary quantum mechanics, Physica, 1946, V.12, 405-460.
  14. Moyal J.E., Quantum mechanics as a statistical theory, Proc. Cambridge Phil. Soc., 1949, V.45, 99-124.
  15. Szabo R.J., Quantum field theory on noncommutative spaces, Phys. Rept., 2003, V.378, 207-299, hep-th/0109162.
  16. Doplicher S., Fredenhagen K., Roberts J.E., Space-time quantization induced by classical gravity, Phys. Lett. B, 1994, V.331, 39-44.
  17. Doplicher S., Spacetime and fields, a quantum texture, hep-th/0105251.
  18. Filk T., Divergencies in a field theory on quantum space, Phys. Lett. B, 1996, V.376, 53-58.
  19. Chen G.H., Wu Y.S., On critical phenomena in a noncommutative space, hep-th/0103020.
  20. Jain A., Joglekar S.D., Causality violation in non-local quantum field theory, Internat. J. Modern Phys. A, 2004, V.19, 3409-3426, hep-th/0307208.
  21. Chaichian M., Kulish P.P., Nishijima K., Tureanu A., On a Lorentz-invariant interpretation of noncommutative space-time and its implications on noncommutative QFT, Phys. Lett. B, 2004, V.604, 98-102, hep-th/0408069.
  22. Minwalla S., Van Raamsdonk M., Seiberg N., Noncommutative perturbative dynamics, JHEP, 2000, N 2, Paper 020, 31 pages, hep-th/9912072.
  23. Hinchliffe I., Kersting N., Ma Y.L., Review of the phenomenology of noncommutative geometry, Internat. J. Modern Phys. A, 2004, V.19, 179-204, hep-ph/0205040.
  24. Gubser S.S., Sondhi S.L., Phase structure of non-commutative scalar field theories, Nuclear Phys. B, 2001, V.605, 395-424, hep-th/0006119.
  25. Castorina P., Zappalà D., Nonuniform symmetry breaking in noncommutative lF4 theory, Phys. Rev. D, 2003, V.68, 065008, 7 pages, hep-th/0303030.
  26. Langmann E., Szabo R.J., Duality in scalar field theory on noncommutative phase spaces, Phys. Lett. B, 2002, V.533, 168-177, hep-th/0202039.
  27. Govindarajan T.R., Kürkçüoglu S., Panero M., Nonlocal regularisation of noncommutative field theories, Modern Phys. Lett. A, 2006, V.21, 1851-1863, hep-th/0604061.
  28. Grosse H., Wulkenhaar R., Renormalisation of f4-theory on noncommutative R4 in the matrix base, Comm. Math. Phys., 2005, V.256, 305-374, hep-th/0401128.
  29. Rivasseau V., Vignes-Tourneret F., Wulkenhaar R., Renormalization of noncommutative f4-theory by multi-scale analysis, Comm. Math. Phys., 2006, V.262, 565-594, hep-th/0501036.
  30. Grosse H., Steinacker H., Renormalization of the noncommutative f3 model through the Kontsevich model, Nuclear Phys. B, 2006, V.746, 202-226, hep-th/0512203.
  31. Grosse H., Steinacker H., A nontrivial solvable noncommutative f3 model in 4 dimensions, hep-th/0603052.
  32. Grosse H., Steinacker H., Exact renormalization of a noncommutative f3 model in 6 dimensions, hep-th/0607235.
  33. Ambjørn J., Makeenko Y.M., Nishimura J., Szabo R.J., Finite N matrix models of noncommutative gauge theory, JHEP, 1999, N 11, Paper 029, 17 pages, hep-th/9911041.
  34. Ambjørn J., Makeenko Y.M., Nishimura J., Szabo R.J., Nonperturbative dynamics of noncommutative gauge theory, Phys. Lett. B, 2000, V.480, 399-408, hep-th/0002158.
  35. Ambjørn J., Makeenko Y.M., Nishimura J., Szabo R.J., Lattice gauge fields and discrete noncommutative Yang-Mills theory, JHEP, 2000, N 5, Paper 023, 49 pages, hep-th/0004147.
  36. Azuma T., Bal S., Nagao K., Nishimura J., Nonperturbative studies of fuzzy spheres in a matrix model with the Chern-Simons term, JHEP, 2004, N 5, Paper 005, 36 pages, hep-th/0401038.
  37. Bietenholz W., Hofheinz F., Nishimura J., The renormalizability of 2D Yang-Mills theory on a non-commutative geometry, JHEP, 2002, N 9, Paper 009, 18 pages, hep-th/0203151.
  38. Bietenholz W., Hofheinz F., Nishimura J., Simulating non-commutative field theory, Nuclear Phys. Proc. Suppl., 2003, V.119, 941-946, hep-lat/0209021.
  39. Ambjørn J., Catterall S., Stripes from (noncommutative) stars, Phys. Lett. B, 2002, V.549, 253-259, hep-lat/0209106.
  40. Bietenholz W., Hofheinz F., Nishimura J., Phase diagram and dispersion relation of the non-commutative lambda f4 model in d = 3, JHEP, 2004, N 6, Paper 042, 36 pages, hep-th/0404020.
  41. Bietenholz W., Bigarini A., Hofheinz F., Nishimura J., Susaki Y., Volkholz J., Numerical results for U(1) gauge theory on 2d and 4d non-commutative spaces, Fortsch. Phys., 2005, V.53, 418-425, hep-th/0501147.
  42. Bietenholz W., Nishimura J., Susaki Y., Volkholz J., A non-perturbative study of 4d U(1) non-commutative gauge theory - the fate of one-loop instability, hep-th/0608072.
  43. Madore J., The fuzzy sphere, Classical Quantum Gravity, 1992, V.9, 69-87.
  44. Alexanian G., Balachandran A.P., Immirzi G., Ydri B., Fuzzy CP2, J. Geom. Phys., 2002, V.42, 28-53, hep-th/0103023.
  45. Balachandran A.P., Dolan B.P., Lee J.H., Martin X., O'Connor D., Fuzzy complex projective spaces and their star-products, J. Geom. Phys., 2002, V.43, 184-204, hep-th/0107099.
  46. Hammou A.B., Lagraa M., Sheikh-Jabbari M.M., Coherent state induced star-product on Rl3 and the fuzzy sphere, Phys. Rev. D, 2002, V.66, 025025, 11 pages, hep-th/0110291.
  47. Medina J., O'Connor D., Scalar field theory on fuzzy S4, JHEP, 2003, N 11, Paper 051, 13 pages, hep-th/0212170.
  48. Vaidya S., Ydri B., On the origin of the UV-IR mixing in noncommutative matrix geometry, Nuclear Phys. B, 2003, V.671, 401-431, hep-th/0305201.
  49. Dolan B.P., O'Connor D., A fuzzy three sphere and fuzzy tori, JHEP, 2003, N 10, Paper 060, 16 pages, hep-th/0306231.
  50. Balachandran A.P., Kürkçüoglu S., Vaidya S., Lectures on fuzzy and fuzzy SUSY physics, hep-th/0511114.
  51. Sheikh-Jabbari M.M., Inherent holography in fuzzy spaces and an N-tropic approach to the cosmological constant problem, hep-th/0605110.
  52. Martin X., A matrix phase for the f4 scalar field on the fuzzy sphere, JHEP, 2004, N 4, Paper 077, 20 pages, hep-th/0402230.
  53. García Floresn F., O'Connor D., Martin X., Simulating the scalar field on the fuzzy sphere, in Proceedings for the XXIIIrd International Symposium on Lattice Field Theory, PoS LAT2005, 2006, 262, 6 pages, hep-lat/0601012.
  54. Medina J., Bietenholz W., Hofheinz F., O'Connor D., Field theory simulations on a fuzzy sphere: an alternative to the lattice, in Proceedings for the XXIIIrd International Symposium on Lattice Field Theory, PoS LAT2005, 2006, 263, 6 pages, hep-lat/0509162.
  55. Azuma T., Bal S., Nagao K., Nishimura J., Absence of a fuzzy S4 phase in the dimensionally reduced 5d Yang-Mills-Chern-Simons model, JHEP, 2004, N 7, Paper 066, 11 pages, hep-th/0405096.
  56. Anagnostopoulos K.N., Azuma T., Nagao K., Nishimura J., Impact of supersymmetry on the nonperturbative dynamics of fuzzy spheres, JHEP, 2005, N 9, Paper 046, 27 pages, hep-th/0506062.
  57. Azuma T., Bal S., Nagao K., Nishimura J., Perturbative versus nonperturbative dynamics of the fuzzy S2 ×S2, JHEP, 2005, N 9, Paper 047, 20 pages, hep-th/0506205.
  58. O'Connor D., Ydri B., Monte Carlo simulation of a NC gauge theory on the fuzzy sphere, hep-lat/0606013.
  59. Panero M., Numerical simulations of a non-commutative theory: the scalar model on the fuzzy sphere, hep-th/0608202.
  60. Grosse H., Klimcík C., Presnajder P., Towards finite quantum field theory in noncommutative geometry, Internat. J. Theor. Phys., 1996, V.35, 231-244, hep-th/9505175.
  61. Chu C.S., Madore J., Steinacker H., Scaling limits of the fuzzy sphere at one loop, JHEP, 2001, N 8, Paper 038, 17 pages, hep-th/0106205.
  62. Dolan B.P., O'Connor D., Presnajder P., Matrix f4 models on the fuzzy sphere and their continuum limits, JHEP, 2002, N 3, Paper 013, 15 pages, hep-th/0109084.
  63. Dolan B.P., O'Connor D., Private communication.
  64. Steinacker H., A non-perturbative approach to non-commutative scalar field theory, JHEP, 2005, N 3, Paper 075, 39 pages, hep-th/0501174.
  65. Klimcík C., Gauge theories on the noncommutative sphere, Comm. Math. Phys., 1998, V.199, 257-279, hep-th/9710153.
  66. Carow-Watamura U., Watamura S., Noncommutative geometry and gauge theory on fuzzy sphere, Comm. Math. Phys., 2000, V.212, 395-413, hep-th/9801195.
  67. Alekseev A.Y., Recknagel A., Schomerus V., Brane dynamics in background fluxes and non-commutative geometry, JHEP, 2000, N 5, Paper 010, 25 pages, hep-th/0003187.
  68. Hashimoto K., Krasnov K., D-brane solutions in non-commutative gauge theory on fuzzy sphere, Phys. Rev. D, 2001, V.64, 046007, 11 pages, hep-th/0101145.
  69. Kimura Y., Noncommutative gauge theories on fuzzy sphere and fuzzy torus from matrix model, Prog. Theoret. Phys., 2001, V.106, 445-469, hep-th/0103192.
  70. Steinacker H., Quantized gauge theory on the fuzzy sphere as random matrix model, Nuclear Phys. B, 2004, V.679, 66-98, hep-th/0307075.
  71. Iso S., Umetsu H., Note on gauge theory on fuzzy supersphere, Phys. Rev. D, 2004, V.69, 105014, 7 pages, hep-th/0312307.
  72. Kimura Y., Nonabelian gauge field and dual description of fuzzy sphere, JHEP, 2004, N 4, Paper 058, 29 pages, hep-th/0402044.
  73. Azuma T., Bal S., Nagao K., Nishimura J., Dynamical aspects of the fuzzy CP2 in the large N reduced model with a cubic term, JHEP, 2006, N 5, Paper 061, 27 pages, hep-th/0405277.
  74. Grosse H., Steinacker H., Finite gauge theory on fuzzy CP2, Nuclear Phys. B, 2005, V.707, 145-198, hep-th/0407089.
  75. Castro-Villarreal P., Delgadillo-Blando R., Ydri B., A gauge-invariant UV-IR mixing and the corresponding phase transition for U(1) fields on the fuzzy sphere, Nuclear Phys. B, 2005, V.704, 111-153, hep-th/0405201.
  76. Behr W., Meyer F., Steinacker H., Gauge theory on fuzzy S2 ×S2 and regularization on noncommutative R4, JHEP, 2005, N 7, Paper 040, 38 pages, hep-th/0503041.
  77. Grosse H., Klimcík C., Presnajder P., Field theory on a supersymmetric lattice, Comm. Math. Phys., 1997, V.185, 155-175, hep-th/9507074.
  78. Grosse H. Reiter G., The fuzzy supersphere, J. Geom. Phys., 1998, V.28, 349-383, math-ph/9804013.
  79. Balachandran A.P., Kürkçüoglu S., Rojas E., The star product on the fuzzy supersphere, JHEP, 2002, N 7, Paper 056, 22 pages, hep-th/0204170.
  80. Balachandran A.P., Pinzul A., Qureshi B., SUSY anomalies break N = 2 to N = 1: the supersphere and the fuzzy supersphere, JHEP, 2005, N 12, Paper 002, 14 pages, hep-th/0506037.
  81. Iso S., Kimura Y., Tanaka K., Wakatsuki K., Noncommutative gauge theory on fuzzy sphere from matrix model, Nuclear Phys. B, 2001, V.604, 121-147, hep-th/0101102.
  82. Iso S., Umetsu H., Gauge theory on noncommutative supersphere from supermatrix model, Phys. Rev. D, 2004, V.69, 105003, 7 pages, hep-th/0311005.
  83. Hasebe K., Kimura Y., Fuzzy supersphere and supermonopole, Nuclear Phys. B, 2005, V.709, 94-114, hep-th/0409230.
  84. Kürkçüoglu S., Non-linear sigma models on the fuzzy supersphere, JHEP, 2004, N 3, Paper 062, 12 pages, hep-th/0311031.
  85. Imai T., Kitazawa Y., Takayama Y., Tomino D., Effective actions of matrix models on homogeneous spaces, Nuclear Phys. B, 2004, V.679, 143-167, hep-th/0307007.
  86. Imai T., Kitazawa Y., Takayama Y., Tomino D., Quantum corrections on fuzzy sphere, Nuclear Phys. B, 2003, V.665, 520-544, hep-th/0303120.
  87. Shimamune Y., On the phase structure of large N matrix models and gauge models, Phys. Lett. B, 1982, V.108, 407-410.
  88. Bleher P., Its A., Double scaling limit in the random matrix model: the Riemann-Hilbert approach, math-ph/0201003.

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