Virasoro 3-algebra from scalar densities

It is shown that the ternary Virasoro-Witt algebra of Curtright, Fairlie and Zachos can be constructed by applying the Nambu commutator to the vect(1) realization on scalar densities. This construction is generalized to vect(d), but the corresponding 3-algebra fails to close. There has recently been a surge of interest in 3-algebras in M-theory [1, 2, 3, 4], which is closely related to the ternary brackets introduced long ago by Nambu [5] and developed by Filippov [6], Curtright and Zachos [7] and many others. In particular, Lin [8] considered a Kac-Moody 3-algebra and Curtright, Fairlie and Zachos [9] considered very recently a 3-algebra related to the Witt (centerless Virasoro) algebra. In this note I observe that their Virasoro-Witt 3-algebra (eqn (22) in [9]) can be constructed by applying the Nambu commutator [A,B,C] = ABC +BCA+ CAB −BAC − CBA−ACB (1) = A[B,C] +B[C,A] + C[A,B] to the Virasoro representation acting on scalar densities, i.e. primary fields. Consider the operators Em = e , Lm = e (−i d dx + λm), (2) Sm = e (−i d dx + λm).