3-form field, we will just consider the bosonic part and ignore coupling to the gauge field, for simplicity. For more details see Refs. [15, 19]. We can start from the Polyakov-type action where γ$_{mn}$ is the auxiliary worldvolume metric and γ = det γ$_{mn}$ . G$_{mn}$ is the induced metric, where m, n = 0 , 1 , 2 and M, N = 0 , 1 , · · · , 10. g$_{MN}$ is the background metric, which will be AdS$_{4}$ × M 1 , 1 , 1 for us in this paper. The variation with respect to γ$_{mn}$ leads to constraint It immediately follows that γ$_{mn}$ = G$_{mn}$ , and if we plug it back to Eq. (8) we arrive at the Nambu-Goto action One can then easily check that P$_{M}$ satisfy the following constraints ( i = 1 , 2 . ) The canonical Hamiltonian is simply zero, thanks to the constraints. In order to switch to Hamiltonian description, we thus need to follow Dirac’s prescription and take a linear combination of the first class primary constraints C$_{m}$ . From the Hamiltonian equation of motion, we have 6 S P = − T$_{2}$ 2 ∫ d $^{3}$σ √ − γ ( γ $^{mn}$G$_{mn}$ − 1) , (8) G$_{mn}$ = ∂$_{m}$X $^{M}$∂$_{n}$ X $^{N}$g$_{MN}$ ( X ) , (9) 2 G$_{mn}$ − γ$_{mn}$γ $^{αβ}$G$_{αβ}$ + γ$_{mn}$ = 0 . (10) S NG = − T$_{2}$ d $^{3}$σ − G. (11) C$_{0}$ ≡ g $^{MN}$P$_{M}$ P$_{N}$ + T 2 $_{2}$GG 00 = 0 , C$_{i}$ ≡ P$_{M}$∂$_{i}$X M = 0 . (13) H = ∫ d $^{2}$σ ( λ $^{0}$C$_{0}$ + λ $^{i}$C$_{i}$ ) . (14) ∂$_{0}$X M = ∂H ∂P$_{M}$ = 2 λ $^{0}$g $^{MN}$P$_{N}$ + λ $^{i}$∂$_{i}$ X M ⇒ P$_{M}$ = 1 2 λ $_{0}$g$_{MN}$ ( ∂$_{0}$ − λ $^{i}$∂$_{i}$ ) X $^{N}$. (15) From the Nambu-Goto action, the generalized momenta are written as P$_{M}$ ( σ ) = − T$_{2}$ − GG 0 $^{n}$∂$_{n}$ X $^{N}$g$_{MN}$ . (12)