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

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

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

The only known compatible solution however is:

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

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]

Bagger, Jonathan; Lambert, Neil (2007-02-26). "Modeling multiple M2-branes". Physical Review D. American Physical Society (APS). 75 (4): 045020. arXiv:hep-th/0611108. doi:10.1103/physrevd.75.045020. ISSN 1550-7998.

Gustavsson, Andreas (2009). "Algebraic structures on parallel M2 branes". Nuclear Physics B. Elsevier BV. 811 (1–2): 66–76. arXiv:0709.1260. doi:10.1016/j.nuclphysb.2008.11.014. ISSN 0550-3213.


Lie 3-Algebra and Multiple M2-branes

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