Morita equivalence of noncommutative supertori. (English) Zbl 1311.81215
Summary: In this paper we study the extension of Morita equivalence of noncommutative tori to the supersymmetric case. The structure of the symmetry group yielding Morita equivalence appears to be intact but its parameter field becomes supersymmetrized having both body and soul parts. Our result is mainly in the two dimensional case in which noncommutative supertori have been constructed recently: The group \(\operatorname{SO}(2,2,\mathcal{V}_Z^0)\), where \(\mathcal{V}_Z^0)\) denotes Grassmann even number whose body part belongs to Z, yields Morita equivalent noncommutative supertori in two dimensions.{
©2010 American Institute of Physics}
©2010 American Institute of Physics}
MSC:
| 81T75 | Noncommutative geometry methods in quantum field theory |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 58B34 | Noncommutative geometry (à la Connes) |
| 15A75 | Exterior algebra, Grassmann algebras |
References:
| [1] | 1.A.Connes, M. R.Douglas, and A.Schwarz, J. High Energy Phys.JHEPFG1126-670802 (1998) 0310.1088/1126-6708/1998/02/003; e-print arXiv:hep-th/9711162. |
| [2] | Polchinski, J., String Theory, 1 and 2 (1998) · Zbl 1006.81521 |
| [3] | Green, M. B.; Scwarz, J. H.; Witten, E., Superstring Theory, 1 and 2 (1987) · Zbl 0619.53002 |
| [4] | 4.M. A.Rieffel and A.Schwarz, Int. J. Math.0129-167X10, 289 (1999)10.1142/S0129167X99000100; e-print arXiv:math.QA/9803057. · Zbl 0968.46060 |
| [5] | 5.A.Schwarz, Nucl. Phys. BNUPBBO0550-3213534, 720 (1998)10.1016/S0550-3213(98)00550-1; e-print arXiv:hep-th/9805034. · Zbl 1079.81066 |
| [6] | 6.N.Berkovits and J.Maldacena, J. High Energy Phys.JHEPFG1126-670809 (2008) 06210.1088/1126-6708/2008/09/062; e-print arXiv:0807.3196. · Zbl 1245.81267 |
| [7] | 7.N.Seiberg, J. High Energy Phys.JHEPFG1126-670806 (2003) 01010.1088/1126-6708/2003/06/010; e-print arXiv:hep-th/0305248; N.Berkovits and N.Seiberg, J. High Energy Phys.JHEPFG1126-670807 (2003) 01010.1088/1126-6708/2003/07/010; e-print arXiv:hep-th/0306226. |
| [8] | Rabin, J. M.; Freund, P. G. O., Commun. Math. Phys., 114, 131 (1988) · Zbl 0648.58008 · doi:10.1007/BF01218292 |
| [9] | Crane, L.; Rabin, J. M., Commun. Math. Phys., 113, 601 (1988) · Zbl 0659.30039 · doi:10.1007/BF01223239 |
| [10] | Rieffel, M., Can. J. Math., 40, 257 (1988) · Zbl 0663.46073 |
| [11] | Manin, Y., e-print arXiv:math.AG/0011197. |
| [12] | Manin, Y., e-print arXiv:math.QA/0307393. |
| [13] | 13.E.Chang-Young and H.Kim, J. Phys. AJPHAC50305-447041, 105201 (2008)10.1088/1751-8113/41/10/105201; e-print arXiv:0709.2483. · Zbl 1189.81105 |
| [14] | Mumford, D., Tata Lectures on Theta III (1991) · Zbl 0744.14033 |
| [15] | Thangavelu, S., Harmonic Analysis on the Heisenberg Group (1998) · Zbl 0892.43001 |
| [16] | Rosenberg, J., Contemp. Math., 365, 123 (2004) |
| [17] | 17.EeC.-Y., H.Kim, and H.Nakajima, J. High Energy Phys.JHEPFG1126-670804 (2008) 00410.1088/1126-6708/2008/04/004; e-print arXiv:0711.4663. · Zbl 1246.81364 |
| [18] | 18.EeC.-Y., H.Kim, and H.Nakajima, J. High Energy Phys.JHEPFG1126-670808 (2008) 05810.1088/1126-6708/2008/08/058; e-print arXiv:0807.0710. |
| [19] | This normalization is different from our previous work (Ref. 18) by a factor of \(2 \pi \). |
| [20] | For the soul part of \(\widetilde{N} \), this can be directly shown by using (43) twice and arbitrariness of \(F^\prime \). |
| [21] | There have been some works on lattice supersymmetry in two different directions: One is to realize lattice supersymmetry by orbifolding of supersymmetric Yang-Mills theory (Ref. 22) and the other is to construct consistent supersymmetry generators directly on lattice (Ref. 23). Although the first approach deals with the lattice formulation of supersymmetric Yang-Mills theory, here we need a noncommutive superlattice formulation in the sense of Seiberg (Ref. 7), not a superbundle on (bosonic) lattice, in order to construct the \(R^p \times Z^q\)-type embedding in the sense of Rieffel (Ref. 10) for noncommutative supertori. Namely, in the embedding of noncommutative \(n\)-tori with \(M = R^p \times Z^q \times F\), where \(2 p + q = n\), the \(Z^q\)-part really deals with the lattice nature of embedding manifold \(M\) while the “finite” \(F\)-part deals with the additional group structure of the bundle living on a torus. The first approach applies the orbifolding method to this \(F\)-part. This type of orbifolding the gauge bundles on noncommutative tori was also investigated earlier in Ref. 24. In the second approach, so far no progress has been reported on noncommutative deformation. We thank M. Unsal for informing us the works done along the first approach. |
| [22] | Unsal, M., J. High Energy Phys., 2005, 12, 33 · doi:10.1088/1126-6708/2005/12/033 |
| [23] | D’Adda, A.; Kanamori, I.; Kawamoto, N.; Nagata, K., Nucl. Phys. B, 798, 168 (2008) · Zbl 1234.81103 |
| [24] | Kim, E.; Kim, H.; Kim, N.; Lee, B. -H.; Lee, C. -Y.; Yang, H. S., Phys. Rev. D, 62, 046001 (2000) · doi:10.1103/PhysRevD.62.046001 |
| [25] | Galperin, A. S.; Ivanov, E. A.; Ogievetsky, V. I.; Sokatchev, E. S., Harmonic Superspace (2001) · Zbl 1029.81003 · doi:10.1017/CBO9780511535109 |
| [26] | Karlhede, A.; Lindstrom, U.; Rocek, M., Phys. Lett. B, 147, 297 (1984) · doi:10.1016/0370-2693(84)90120-5 |
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