Classical equivalence of Polyakov and Nambu–Goto actions ($h_{ab}={\Omega }^2{\gamma }_{ab}$)

Varying the independent metric in the Polyakov action sets the worldsheet stress tensor to zero. For a nondegenerate timelike embedding in two worldsheet dimensions, the metric is proportional to the induced worldsheet metric. The conformal factor cancels from the action, which becomes the Nambu–Goto action. This elimination is classical; the quantum measure and worldsheet Weyl anomaly require separate treatment.