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And for polar angles µ, ˜ θ and θ , we assume they do not have τ -dependence, like µ ( σ$_{1}$, σ$_{2}$ ). 

With this choice, we can readily write down several constants of motion. They are momenta conjugate to those coordinates with linear τ -dependence, and it is natural to call them E, J$_{ψ}$, J$_{˜}$ $_{ψ}$, J$_{˜}$ $_{φ}$, J$_{φ}$ . It is easy to infer that E gives conformal dimension, J$_{ψ}$ gives R- charge, and J$_{φ}$ gives the spin quantum number of SU (2) global symmetry, for the dual operators in the field theory. 

Maybe it is not immediately clear how the SU (3) angular momenta J$_{˜}$ ψ and J$_{˜}$ φ give the weight vector of a given state. Let us follow the the standard convention and define the Cartan subalgebra with Gell-mann matrices 

<img src='content_image/352979.jpg'>

for the fundamental representation. Then it turns out that we can relate J$_{˜}$ φ with λ$_{3}$ , and J$_{˜}$ ψ − 2 J$_{ψ}$ with λ$_{8}$ . This adjustment will be justified when we study particle-like solutions. 

Below we record the expressions for various conserved quantities of spinning membrane configurations. We have introduced new symbols Δ ≡ E, R ≡ 8 J$_{ψ}$, Q$_{3}$ ≡ 2 J$_{˜}$ $_{φ}$, Q$_{8}$ ≡ 2 √ 3( J$_{˜}$ ψ − 2 J$_{ψ}$ ) , J$_{3}$ ≡ 2 J$_{φ}$ for later convenience, and √ λ ′ = L $^{2}$/ 2 λ $^{0}$. 

<img src='content_image/352980.jpg'>

