Polyakov action
= Polyakov action
{c}
{wiki}
The Polyakov action introduces an independent worldsheet metric $g_{\alpha\beta}$:
$$
S_{\rm P}=-\frac T2\int_\Sigma d^2\sigma\,\sqrt{-g}\,
g^{\alpha\beta}\partial_\alpha X\mathbin\cdot\partial_\beta X.
$$
Its metric equation makes $g_{\alpha\beta}$ conformal to the <induced worldsheet metric>, reducing it classically to the <Nambu–Goto action>.