Gauge Fixing

Locally, we set the world-sheet metric to any convenient constant form by a gauge transformation. The simplest choice is

$h_{ab}={\eta }_{ab}:=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$.

The action then simplifies to

$S=-\frac{T}{2}{\int }_{\Sigma }{d}^2\sigma {\eta }^{ab}{\partial }_a{X}^{\mu }{\partial }_b{X}_{\mu }=-\frac{T}{2}{\int }_{\Sigma }{d}^2\sigma ({\partial }_0{X}^{\mu }{\partial }_0{X}_{\mu }-{\partial }_1{X}^{\mu }{\partial }_1{X}_{\mu })$.

In this gauge, the coordinate functions X${}^{\mu }(\sigma )$ are free fields in two dimensions.

The equation of motion is then the wave equation $({\partial }_0^2-{\partial }_1^2)X^{\mu }=0$ with a general solution

$X^{\mu }(\sigma )=X_L^{\mu }({\sigma }^0+{\sigma }^1)+X_R^{\mu }({\sigma }^0-{\sigma }^1)$

for arbitrary functions X${}_L$ and X${}_R$. The subscripts refer to the left-moving and right-moving wave solutions respectively.

We now introduce light-cone coordinates as ${\sigma }^{\pm }:={\sigma }^0\pm {\sigma }^1$, where the metric components are

${\eta }_{+-}={\eta }_{-+}=-\frac{1}{2};{\eta }_{++}={\eta }_{--}=0$

${\eta }^{+-}={\eta }^{-+}=-2;{\eta }^{++}={\eta }^{--}=0$

The partial derivatives are given by

${\partial }_{\pm }=\frac{1}{2}({\partial }_0\pm {\partial }_1)$ and the wave operator is ${\partial }_0^2-{\partial }_1^2=4{\partial }_{+}{\partial }_{-}$.

The gauge-fixed Polyakov action becomes

$S=T{\int }_{\Sigma }{d}^2\sigma {\partial }_{+}{X}^{\mu }{\partial }_{-}{X}_{\mu }$.

The equations of motion for the world-sheet metric components $h_{ab}(\mathbf{\sigma })$ are $T_{ab}=0$.

In light-cone coordinates, the components of the stress-energy tensor are

$T_{++}={\partial }_{+}{X}^{\mu }{\partial }_{+}{X}_{\mu }{T}_{--}={\partial }_{-}{X}^{\mu }{\partial }_{-}{X}_{\mu }{T}_{+-}=T_{-+}=0$.