Bagger–Lambert–Gustavsson action

In theoretical physics, in the context of M-theory, the action for the N =8 M2 branes in full is (with some indices hidden):

${\displaystyle S=\int {\left(-{\frac {1}{2}}D^{\mu }X_{I}D_{\mu }X_{I}+{\frac {i}{2}}{\overline {\Psi }}\Gamma ^{\mu }D_{\mu }\Psi +{\frac {i}{4}}{\overline {\Psi }}\Gamma _{IJ}\left[X^{I},X^{J},\Psi \right]-{\frac {1}{12}}\left[X^{I},X^{J},X^{K}\right]\left[X^{I},X^{J},X^{K}\right]+{\frac {1}{2}}\varepsilon ^{abc}Tr(A_{a}\partial _{b}A_{c}+{\frac {2}{3}}A_{a}A_{b}A_{c})\right)}d\sigma ^{3}}$

where [, ] is a generalisation of a Lie bracket which gives the group constants.

The only known compatible solution however is:

${\displaystyle \left[A,B,C\right]_{\eta }\equiv \varepsilon ^{\mu \nu \tau \eta }A_{\mu }B_{\nu }C_{\tau }}$

using the Levi-Civita symbol which is invariant under SO(4) rotations. M5 branes can be introduced by using an infinite symmetry group.

The action is named after Jonathan Bagger, Neil Lambert, and Andreas Gustavsson. ^{[1] [2] [3]}

Notes

1. Bagger, Jonathan; Lambert, Neil (2007-02-26). "Modeling multiple M2-branes". Physical Review D. 75 (4) 045020. arXiv: hep-th/0611108 . Bibcode:2007PhRvD..75d5020B. doi:10.1103/physrevd.75.045020. ISSN   1550-7998. S2CID   119483842.
2. Gustavsson, Andreas (2009). "Algebraic structures on parallel M2 branes". Nuclear Physics B. 811 (1–2): 66–76. arXiv: 0709.1260 . Bibcode:2009NuPhB.811...66G. doi:10.1016/j.nuclphysb.2008.11.014. ISSN   0550-3213. S2CID   8856345.
3. Bagger, Jonathan; Lambert, Neil (2008-03-07). "Gauge Symmetry and Supersymmetry of Multiple M2-branes". Physical Review D. 77 (6) 065008. arXiv: 0711.0955 . Bibcode:2008PhRvD..77f5008B. doi:10.1103/PhysRevD.77.065008. S2CID   14988717.

References

• Lie 3-Algebra and Multiple M2-branes