On the vacuum states for non-commutative gauge theory. (English) Zbl 1189.81214
Summary: Candidates for renormalizable gauge theory models on Moyal spaces constructed recently have non-trivial vacua. We show that these models support vacuum states that are invariant under both global rotations and symplectic isomorphisms which form a global symmetry group for the action. We compute the explicit expression in position space for these vacuum configurations in two and four dimensions.
MSC:
| 81T75 | Noncommutative geometry methods in quantum field theory |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
References:
| [1] | M.R. Douglas, N.A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001) [hep-th/0106048] · Zbl 1205.81126 · doi:10.1103/RevModPhys.73.977 |
| [2] | R.J. Szabo, Phys. Rep. 378, 207 (2003) [hep-th/0109162] · Zbl 1042.81581 · doi:10.1016/S0370-1573(03)00059-0 |
| [3] | R. Wulkenhaar, J. Geom. Phys. 56, 108 (2006) · Zbl 1083.81014 · doi:10.1016/j.geomphys.2005.04.019 |
| [4] | A. Connes, Non-commutative Geometry (Academic Press Inc., San Diego, 1994), available at http://www.alainconnes.org/downloads.html |
| [5] | A. Connes, M. Marcolli, A walk in the non-commutative garden (2006), available at http://www.alainconnes.org/downloads.html · Zbl 1145.14005 |
| [6] | N. Seiberg, E. Witten, JHEP 9909, 032 (1999) [hep-th/9908142] · Zbl 0957.81085 · doi:10.1088/1126-6708/1999/09/032 |
| [7] | V. Schomerus, JHEP 9906, 030 (1999) [hep-th/9903205] · Zbl 0961.81066 · doi:10.1088/1126-6708/1999/06/030 |
| [8] | J.M. Gracia-Bondía, J.C. Várilly, J. Math. Phys. 29, 869 (1988) · Zbl 0652.46026 · doi:10.1063/1.528200 |
| [9] | J.C. Várilly, J.M. Gracia-Bondía, J. Math. Phys. 29, 880 (1988) · Zbl 0652.46027 · doi:10.1063/1.527984 |
| [10] | L. Susskind, The Quantum Hall Fluid and Non-Commutative Chern Simons Theory, hep-th/0101029 |
| [11] | S. Hellerman, M. Van Raamsdonk, JHEP 10, 39 (2001) · doi:10.1088/1126-6708/2001/10/039 |
| [12] | S. Minwalla, M. Van Raamsdonk, N. Seiberg, JHEP 0002, 020 (2000) [hep-th/9912072] · Zbl 0959.81108 · doi:10.1088/1126-6708/2000/02/020 |
| [13] | I. Chepelev, R. Roiban, JHEP 0005, 037 (2000) [hep-th/9911098] · Zbl 0990.81756 · doi:10.1088/1126-6708/2000/05/037 |
| [14] | A. Matusis, L. Susskind, N. Toumbas, JHEP 0012, 002 (2000) [hep-th/0002075] · Zbl 0990.81549 · doi:10.1088/1126-6708/2000/12/002 |
| [15] | H. Grosse, R. Wulkenhaar, Commun. Math. Phys. 256, 305 (2005) [hep-th/0401128] · Zbl 1075.82005 · doi:10.1007/s00220-004-1285-2 |
| [16] | H. Grosse, R. Wulkenhaar, Commun. Math. Phys. 254, 91 (2005) [hep-th/0305066] · Zbl 1079.81049 · doi:10.1007/s00220-004-1238-9 |
| [17] | V. Rivasseau, F. Vignes-Tourneret, R. Wulkenhaar, Commun. Math. Phys. 262, 565 (2006) [hep-th/0501036] · Zbl 1109.81056 · doi:10.1007/s00220-005-1440-4 |
| [18] | R. Gurau, J. Magnen, V. Rivasseau, F. Vignes-Tourneret, Commun. Math. Phys. 267, 515 (2006) [hep-th/0512271] · Zbl 1113.81101 · doi:10.1007/s00220-006-0055-8 |
| [19] | R. Gurau, A. Tanasa, arXiv:0706.1147[math-ph] |
| [20] | V. Rivasseau, Non-commutative renormalization, arXiv:0705.0705 [hep-th] · Zbl 1166.81363 |
| [21] | J.-C. Wallet, J. Phys.: Conf. Ser. 103, 012007 (2008) [arXiv:0708.2471 [hep-th]] · doi:10.1088/1742-6596/103/1/012007 |
| [22] | E. Langmann, R.J. Szabo, Phys. Lett. B 533, 168 (2002) [hep-th/0202039] · Zbl 0994.81116 · doi:10.1016/S0370-2693(02)01650-7 |
| [23] | H. Grosse, R. Wulkenhaar, JHEP 0312, 019 (2003) [hep-th/0307017] · doi:10.1088/1126-6708/2003/12/019 |
| [24] | E. Langmann, R.J. Szabo, K. Zarembo, JHEP 0401, 017 (2004) [hep-th/0308043] · Zbl 1243.81205 · doi:10.1088/1126-6708/2004/01/017 |
| [25] | E. Langmann, R.J. Szabo, K. Zarembo, Phys. Lett. B 569, 95 (2003) [hep-th/0303082] · Zbl 1059.81608 · doi:10.1016/j.physletb.2003.07.020 |
| [26] | F. Vignes-Tourneret, Ann. Inst. Henri Poincare 8, 427 (2007) [math-ph/0606069] · Zbl 1133.81058 · doi:10.1007/s00023-006-0312-6 |
| [27] | H. Grosse, R. Wulkenhaar, Eur. Phys. J. C 35, 277 (2004) [hep-th/0402093] · Zbl 1191.81207 · doi:10.1140/epjc/s2004-01853-x |
| [28] | A. Lakhoua, F. Vignes-Tourneret, J.-C. Wallet, Eur. Phys. J. C 52, 735 (2007) [hep-th/0701170] · Zbl 1189.81219 · doi:10.1140/epjc/s10052-007-0424-2 |
| [29] | M. Disertori, V. Rivasseau, Eur. Phys. J. C 50, 661 (2007) [hep-th/0610224] · Zbl 1248.81254 · doi:10.1140/epjc/s10052-007-0211-0 |
| [30] | M. Disertori, R. Gurau, J. Magnen, V. Rivasseau, Phys. Lett. B 649, 95 (2007) [hep-th/0612251] · Zbl 1248.81253 · doi:10.1016/j.physletb.2007.04.007 |
| [31] | R. Gurau, J. Magnen, V. Rivasseau, A. Tanasa, A translation-invariant renormalizable non-commutative scalar model, arXiv:0802.0791 [hep-th] · Zbl 1170.81041 |
| [32] | A. de Goursac, J.-C. Wallet, R. Wulkenhaar, Eur. Phys. J. C 51, 977 (2007) [hep-th/0703075] · Zbl 1189.81215 · doi:10.1140/epjc/s10052-007-0335-2 |
| [33] | H. Grosse, M. Wohlgenannt, Eur. Phys. J. C 52, 435 (2007) [hep-th/0703169] · Zbl 1189.81217 · doi:10.1140/epjc/s10052-007-0369-5 |
| [34] | A. de Goursac, J. Phys.: Conf. Ser. 103, 012010 (2008) [arXiv:0710.1162 [hep-th]] · doi:10.1088/1742-6596/103/1/012010 |
| [35] | H. Grosse, M. Wohlgennant, J. Phys.: Conf. Ser. 53, 764 (2006) [hep-th/0607208] · doi:10.1088/1742-6596/53/1/050 |
| [36] | M. Dubois-Violette, R. Kerner, J. Madore, J. Math. Phys. 31, 323 (1990) · Zbl 0704.53082 · doi:10.1063/1.528917 |
| [37] | T. Masson, J. Geom. Phys. 31, 142 (1999) · Zbl 0940.46046 · doi:10.1016/S0393-0440(99)00009-1 |
| [38] | H. Grosse, R. Wulkenhaar, 8D-spectral triple on 4D-Moyal space and the vacuum of non-commutative gauge theory, arXiv:0709.0095 [hep-th] · Zbl 1243.58005 |
| [39] | A. de Goursac, A. Tanasa, J.-C. Wallet, Eur. Phys. J. C 53, 459 (2008) [arXiv:0709.3950 [hep-th]] · Zbl 1189.81213 · doi:10.1140/epjc/s10052-007-0465-6 |
| [40] | H. Weyl, The Classical Groups (Princeton University Press, Princeton, 1946) · Zbl 1024.20502 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
