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Triality and Bagger-Lambert Theory

We present two alternative field contents for Bagger-Lambert theory, based on the triality of SO(8). The first content is (φ_{A a}, χ_{\dot A a} ; A_\m{}^{a b}), where the bosonic field φis in the 8_S of SO(8) instead of the 8_V as in the original Bagger-Lambert formulation. The second field content is (φ_{\dot A a}, χ^I{}_a ; A_\m{}^{a b}), where the bosonic field φand the fermionic field χare respectively in the 8_C and 8_V of SO(8). In both of these field contents, the bosonic potentials are positive definite, as desired. Moreover, these bosonic potentials can be unified by the triality of SO(8). To this end, we see a special constant matrix as a product of two SO(8) generators playing an important role, relating the 8_V, 8_S and 8_C of SO(8) for the triality. As an important application, we give the supersymmetry transformation rule for N=6 superconformal Chern-Simons theory with the supersymmetry parameter in the 6 of SO(6), obtained by the truncation of our first field content.