content_image/354168.jpg content_image/354169.jpg content_image/354170.jpg content_image/354171.jpg content_image/354176.jpg content_image/354172.jpg content_image/354174.jpg content_image/354173.jpg
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 

<img src='content_image/354168.jpg'>

where γ$_{mn}$ is the auxiliary worldvolume metric and γ = det γ$_{mn}$ . G$_{mn}$ is the induced metric, 

<img src='content_image/354169.jpg'>

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 

<img src='content_image/354170.jpg'>

It immediately follows that γ$_{mn}$ = G$_{mn}$ , and if we plug it back to Eq. (8) we arrive at the Nambu-Goto action 

<img src='content_image/354171.jpg'>

From the Nambu-Goto action, the generalized momenta are written as 

<img src='content_image/354176.jpg'>

One can then easily check that P$_{M}$ satisfy the following constraints ( i = 1 , 2 . ) 

<img src='content_image/354172.jpg'>

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 

<img src='content_image/354174.jpg'>

<img src='content_image/354173.jpg'>

