The polyakov action is $$ S=-\frac{T}{2}\int d\tau d\sigma \sqrt{-h}h^{ab}\partial_ax^\mu\partial_b x^\nu g_{\mu\nu} $$ where $g$ is the background spacetime metric and $h$ is worldsheet metric. I've seen on many notes on string theory that from the reparametrization invariance of this action, and the weyl symmetry of $h$, we can make $h=\eta$, the flat metric (on worldsheet). But I'm wondering is there some explicit derivation, start from an arbitrary $h$, which satisfies the equation of motion for $h$, i.e. $$ \partial_a x^\mu \partial_b x^\nu g_{\mu\nu}=\frac{1}{2}h_{ab}h^{cd}\partial_cx^\mu\partial_d x^\nu g_{\mu\nu} $$ then how can we prove that $h$ can be written as a flat metric? More concretely, I wonder how can we use reparametrization invariance to write $h$ as $h=e^{\phi(\tau,\sigma)}\eta$ for some function $\phi$. If so, then Weyl invariance implies flatness directly. But reparametrization invariance is a symmetry of the action, not for $h$ itself, I don't see how reparametrization works here.
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