Why can't the dynamical metric in the Polyakov Action be the induced metric?

Question

The Polyakov action is given by
$$
S_{P} = -frac{T}{2} int d^2sigma sqrt{h} h^{mu nu} gamma_{mu nu}
tag{1}$$
where $h_{mu nu}$ is the dynamical metric and $gamma_{mu nu}$ is the induced metric. It is well known that this is equivalent to the Nambu-Goto Action
$$
S_{NG} = -T int d^2sigma sqrt{gamma}.tag{2}
$$
Theoretically, I don't see any reason why the dynamical metric cannot be the same as the induced metric. However, if the dynamical metric is set equal to the induced metric in $S_P$, the two actions differ by a factor of $frac{1}{2}$. Am I correct in that this means that the dynamical metric cannot be the induced metric? If so, why?

Nogueira · Answer

Your factors of $2$ are wrong. You do not get the $1/2$ difference because
$$
gamma^{munu}gamma_{munu}=delta_{mu}^{mu}=2
$$
So replacing $h^{munu}$ by $gamma^{munu} $ in eq. 1 gives eq. 2.
Note that you can also replace $h^{munu}$ by $e^{-2Omega(sigma)}gamma^{munu}$ in eq. 1 and still get eq. 2. This is so because $sqrt{h}= e^{+2Omega(sigma)}sqrt{gamma}$ if $h^{munu}=e^{-2Omega(sigma)}gamma^{munu}$. This is the Weyl symmetry of the Polyakov action, i.e. $h_{munu} $ is only defined up to a scaling factor.

Why can't the dynamical metric in the Polyakov Action be the induced metric?

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