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): S = ∫ ( − 1 2 D μ X I D μ X I + i 2 Ψ ¯ Γ μ D μ Ψ + i 4 Ψ ¯ Γ I J [ X I , X J , Ψ ] − 1 12 [ X I , X J , X K ] [ X I , X J , X K ] + 1 2 ε a b c T r ( A a ∂ b A c + 2 3 A a A b A c ) ) d σ 3 {\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: [ A , B , C ] η ≡ ε μ ν τ η A μ B ν C τ {\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.

Source: Wikipedia — Bagger–Lambert–Gustavsson action (CC BY-SA 4.0)

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): S = ∫ ( − 1 2 D μ X I D μ X I + i 2 Ψ ¯ Γ μ D μ Ψ + i 4 Ψ ¯ Γ I J [ X I , X J , Ψ ] − 1 12 [ X I , X J , X K ] [ X I , X J , X K ] + 1 2 ε a b c T r ( A a ∂ b A c + 2 3 A a A b A c ) ) d σ 3 {\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: [ A , B , C ] η ≡ ε μ ν τ η A μ B ν C τ {\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.

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Source: Wikipedia "Bagger–Lambert–Gustavsson action" · CC BY-SA 4.0

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