 Phase-space identification of the Polyakov metric

For positive lapse, eliminating momentum gives $(\dot{X}-u{X}^{\prime }{)}^2/(2e)-e{T}^2{X}^{\prime 2}/2$. This is the Polyakov action for ${\gamma }_{ab}=\left(\begin{array}{cc}{u}^2-e^2{T}^2& u\\ u& 1\end{array}\right)$, up to a Weyl transformation. Thus $e=1/T,u=0$ is conformal gauge.