Consider a 2d Conformal Field Theory, with the metric of the underlying spacetime being $\gamma_{ab}$. I understand that we have the freedom to set $\gamma_{ab}$ to a flat form (either Euclidean or Minkowski).
As far as I know, this can be done only in 2 dimensions due to the coordinate diffeomorphisms and Weyl invariance of the respective field theory on the 2d spacetime. It is what we do in the context of String Theory, where we fix a part of the gauge freedom by setting $\gamma_{ab}=\eta_{ab}$ ($\gamma_{ab}$ being the 2d worldsheet metric).
Is there an obvious connection between the conformal invariance of the 2d spacetime and the diffeo - Weyl invariance of the field theory defined on the 2d spacetime?
