Lie 3-algebra and super-affinization of split-octonions. (English) Zbl 1274.81211
Summary: The purpose of this study is to extend the concept of a generalized Lie 3-algebra, known to the divisional algebra of the octonions \(\mathbb O\), to split-octonions \(\mathbb {SO}\), which is non-divisional. This is achieved through the unification of the product of both of the algebras in a single operation. Accordingly, a notational device is introduced to unify the product of both algebras. We verify that \(\mathbb{SO}\) is a Malcev algebra and we recalculate known relations for the structure constants in terms of the introduced structure tensor. Finally we construct the manifestly supersymmetric \(\mathcal N=1\mathbb{SO}\) affine superalgebra. An application of the split Lie 3-algebra for a Bagger and Lambert gauge theory is also discussed.
MSC:
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 81R15 | Operator algebra methods applied to problems in quantum theory |
| 11R52 | Quaternion and other division algebras: arithmetic, zeta functions |
| 17A35 | Nonassociative division algebras |
| 17A70 | Superalgebras |
References:
| [1] | DOI: 10.1090/S0273-0979-01-00934-X · Zbl 1026.17001 · doi:10.1090/S0273-0979-01-00934-X |
| [2] | DOI: 10.1103/PhysRevD.75.045020 · doi:10.1103/PhysRevD.75.045020 |
| [3] | Bagger J., JHEP 02 pp 105– |
| [4] | DOI: 10.1103/PhysRevD.77.065008 · doi:10.1103/PhysRevD.77.065008 |
| [5] | DOI: 10.1103/PhysRevD.79.025002 · Zbl 1222.81264 · doi:10.1103/PhysRevD.79.025002 |
| [6] | Benichou R., JHEP 1004 pp 121– |
| [7] | DOI: 10.1088/0305-4470/36/13/312 · Zbl 1040.37064 · doi:10.1088/0305-4470/36/13/312 |
| [8] | DOI: 10.1103/PhysRevD.78.066019 · doi:10.1103/PhysRevD.78.066019 |
| [9] | DOI: 10.1063/1.532950 · Zbl 0969.81004 · doi:10.1063/1.532950 |
| [10] | Duff M. J., Lect. Notes Phys. 755 pp 93– |
| [11] | DOI: 10.1007/BF00402262 · Zbl 0692.22005 · doi:10.1007/BF00402262 |
| [12] | DOI: 10.1016/0550-3213(96)00088-0 · Zbl 1002.81507 · doi:10.1016/0550-3213(96)00088-0 |
| [13] | Gustavsson A., JHEP 04 pp 083– |
| [14] | DOI: 10.1016/j.nuclphysb.2008.11.014 · Zbl 1194.81205 · doi:10.1016/j.nuclphysb.2008.11.014 |
| [15] | DOI: 10.1016/S0375-9601(01)00709-5 · Zbl 1036.81016 · doi:10.1016/S0375-9601(01)00709-5 |
| [16] | III W. D. L., JHEP 0911 pp 007– |
| [17] | Gomis G. M. J., JHEP 06 pp 075– |
| [18] | DOI: 10.1007/978-1-4612-0503-6 · doi:10.1007/978-1-4612-0503-6 |
| [19] | Krishnan C., JHEP 07 pp 005– |
| [20] | DOI: 10.1016/0550-3213(83)90584-9 · doi:10.1016/0550-3213(83)90584-9 |
| [21] | Lee K., JHEP 04 pp 012– |
| [22] | Lin H., JHEP 0807 pp 136– |
| [23] | Lust D., Lecture Notes in Physics 346, in: Lectures on String Theory (1989) |
| [24] | DOI: 10.4310/ATMP.1998.v2.n2.a1 · Zbl 0914.53047 · doi:10.4310/ATMP.1998.v2.n2.a1 |
| [25] | DOI: 10.1103/PhysRevD.7.2405 · Zbl 1027.70503 · doi:10.1103/PhysRevD.7.2405 |
| [26] | Aharony D. L. J. O., JHEP 10 pp 091– |
| [27] | de Medeiros J. M. F.-O. P., JHEP 07 pp 111– |
| [28] | de Medeiros J. M. F.-O. P., JHEP 08 pp 045– |
| [29] | DOI: 10.1088/1751-8113/43/1/015205 · Zbl 1180.81116 · doi:10.1088/1751-8113/43/1/015205 |
| [30] | Sheikh-Jabbari M. M., JHEP 12 pp 111– |
| [31] | Terashima S., JHEP 12 pp 059– |
| [32] | DOI: 10.4310/ATMP.1998.v2.n2.a2 · Zbl 0914.53048 · doi:10.4310/ATMP.1998.v2.n2.a2 |
| [33] | DOI: 10.1016/j.physletb.2008.11.001 · doi:10.1016/j.physletb.2008.11.001 |
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