... 1
E-mail address: ansoldi@trieste.infn.it 
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... Castro2
E-mail address: castro@ctsps.cau.edu 
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...E.Spallucci3
E-mail address: spallucci@vstst0.ts.infn.it 
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... understood4
There are many interesting reviews on this problem, as [1], [2], [3], ...and we apologize for omitting many other good ones. 
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... respectively5
The metric signature is $(- \, + \, + \, +) $ and in our notation matrices are denoted by boldface letters. 
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... field6
For the sake of clarity, let us summarize the canonical dimensions in natural units of various quantities:
$\displaystyle [ A_\mu{}^a(x) ]$ $\textstyle \equiv$ $\displaystyle [ \mbox{\textit{\bf{}A}}_\mu^{(\mathrm{q})} ] = ( \mathrm{length} )^{-1}$  
$\displaystyle [ F_{\mu\nu}{}^a(x) ]$ $\textstyle \equiv$ $\displaystyle [ \mbox{\textit{\bf{}F}}_{\mu\nu}{}^{(\mathrm{q})} ]=( \mathrm{length} )^{-2}$  
$\displaystyle [ \sigma^m ]$ $\textstyle =$ $\displaystyle ( \mathrm{length} )^0=1\quad,\quad[ t ]= \mathrm{length}\quad, \quad [ \beta ]= \mathrm{length}$  
$\displaystyle [ g_0 ]$ $\textstyle \equiv$ $\displaystyle [ g_{\mathrm{YM}} ]=( \mathrm{length} )^0=1$  
$\displaystyle [ V_H ]$ $\textstyle =$ $\displaystyle ( \mathrm{length} )^3$  
$\displaystyle [ X^i ]$ $\textstyle =$ $\displaystyle \mathrm{length}$  
$\displaystyle [ \mu_0 ]$ $\textstyle =$ $\displaystyle ( \mathrm{length} )^{-1}\quad,\quad [ \alpha ]= ( \mathrm{length} )^{-5} \ .$   

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... surface7
We have assumed that $X^i$ are three spacelike coordinates. By relaxing this assumptions we can give (38) a slightly different physical interpretation. If all three $X^i$ are considered as transverse directions, then (38) can be seen as the light-cone gauge action for a bosonic brane in a $5$-dimensional target spacetime. 
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