The Polyakov Action

Since there is a square root in the Nambu-Goto action, quantisation is difficult. So, we switch to a classically equivalent action that is quadratic in $X^{\mu }(\sigma )$.

We introduce an additional field $h_{ab}(\sigma )$ describing the world-sheet gravity.

The new action principle: $S=-\frac{T}{2}{\int }_{\Sigma }{d}^2\sigma \sqrt{-h}{h}^{ab}{\gamma }_{ab}=-\frac{T}{2}{\int }_{\Sigma }{d}^2\sigma \sqrt{-h}{h}^{ab}{\partial }_a{X}^{\mu }{\partial }_b{X}^{\nu }{\eta }_{\mu \nu }$.

Here, $h_{ab}(\sigma )$ is the intrinsic metric on the world-sheet with inverse $h^{ab}(\sigma )$ and $h:=\det (h_{ab})<0$ is the determinant of the intrinsic world-sheet metric. This is known as the Polyakov Action.

The Nambu-Goto and Polyakov actions are classically equivalent. How?

The equations of motion are $0=\frac{\delta S}{\delta h^{ab}}=-\frac{T}{2}\sqrt{-h}{T}_{ab}$

where $T_{ab}:=-\frac{2}{T}\frac{1}{\sqrt{-h}}\frac{\delta S}{(}\delta h^{ab})$, known as the world-sheet energy-momentum tensor.

Note: $T_{ab}={\partial }_a{X}^{\mu }{\partial }_b{X}_{\mu }-\frac{1}{2}{h}_{ab}{h}^{cd}{\partial }_c{X}^{\mu }{\partial }_d{X}_{\mu }$.

Since the world-sheet metric must be non-degenerate, $\sqrt{-h}\ne 0$, and the equation of motion is simply $T_{ab}=0$.

This can be expressed as

${\gamma }_{ab}=\frac{1}{2}(h^{cd}{\gamma }_{cd})h_{ab}$

which implies that intrinsic world metric and the induced world metric are proportional. Taking the determinant,

$\det (\gamma )=\frac{1}{4}(h^{cd}{\gamma }_{cd}{)}^2\det (h)$, which produces the Nambu-Goto action when substituted back in.