Light-cone Gauge

Singling out one component of $X^{\mu }$ will manifestly break D-dimensional Poincaré invariance. Let us introduce spacetime light-cone coordinates:

$X^{\pm }=\frac{1}{\sqrt{2}}(X^0\pm X^1)$.

The remaining transverse coordinates are denoted by $X^i=1,...,D-2$. In these coordinates, indices are raised and lowered by

$V^{+}=-V_{-}{V}^{-}=-V_{+}{V}^i=V_i$

and

$U.V=U^i{V}_i-U^{+}{V}^{-}-U^{-}{V}^{+}$.

The light-cone gauge condition singles out the components $X^{+}$.

Consider the expansion of $X^{\mu }(\tau ,\sigma )$ for the classical open string:

$X^{\mu }(\tau ,\sigma )=x^{\mu }+{\ell }^2{p}^{\mu }\tau +i\ell {\sum }_{n\ne 0}\frac{1}{n}{\alpha }_n^{\mu }{e}^{-in\tau }\cos (n\sigma )$.

The light-cone gauge condition uses the residual gauge transformation to choose the world-sheet time coordinate $\tau =X^{+}/p^{+}+\text{constant}$, or more precisely

$X^{+}(\tau ,\sigma )=x^{+}+{\ell }^2{p}^{\mu }\tau$.

This condition sets the modes ${\alpha }_n^{+}=0$ for all $n\ne 0$. Let us show that the Virasoro constraints can be solved and $X^{-}$ coordinate function be eliminated, leaving only the transverse coordinates $X^i$. Recall,

$(\dot{X}\pm X^{\prime }{)}^2=0$.

In light-cone gauge,

${\dot{X}}^{+}={\ell }^2{p}^{+}{X}^{\prime +}=0$.

Thus, our constraints become, ${\sum }_i({\dot{X}}^i\pm X^{\prime i}{)}^2=2{\ell }^2{p}^{+}({\dot{X}}^{-}\pm X^{\prime -})$ where $i=2,\dots ,D-1$. With the exception of the constant term $x^{-}$, these two equations allow us to completely solve for $X^{-}$ in terms of the $X^i$ and so, we no longer have independent oscillator modes in this direction either.

Let us now express the constraints in terms of the Fourier modes as $L_m=0$ for all integers $m$. Thus, expanding all the Virasoro generators in light-cone gauge,

$L_m=\frac{1}{2}{\sum }_{n=-\infty }^{\infty }({\alpha }_{m-n}^i{\alpha }_n^i-{\alpha }_{m-n}^{-}{\alpha }_n^{-}-{\alpha }_{m-n}^{-}{\alpha }_n^{+})=\frac{1}{2}{\sum }_{n=-\infty }^{\infty }{\alpha }_{m-n}^i{\alpha }_n^i-\frac{1}{2}({\alpha }_0^{+}{\alpha }_m^{-}-{\alpha }_m^{-}{\alpha }_0^{+})$

$=\frac{1}{2}{\sum }_{n=-\infty }^{\infty }{\alpha }_{m-n}^i{\alpha }_n^i-\ell p^{+}{\alpha }_m^{-}$.

The constraints $L_m=0$ are therefore solved by ${\alpha }_m^{-}=\frac{1}{2\ell p^{+}}{\sum }_{n=-\infty }^{\infty }{\alpha }_{m-n}^i{\alpha }_n^i$ and this eliminates the modes ${\alpha }_m^{-}$ in terms of the light-come momentum $p^{+}$ and the transverse modes ${\alpha }_m^i$. The special case $L_0=0$ reproduces the mass-shell condition

${\alpha }^{\prime }{M}^2=N$ where $N={\sum }_{m=1}^{\infty }{\alpha }_{-m}^i{\alpha }_m^i$ is the classical number operator but now restricted to only the transverse modes.

Similarly, for the closed string, we obtain the classical mass-shell and level-matching conditions

${\alpha }^{\prime }{M}^2=2(N+\tilde{N})$ and $N-\tilde{N}=0$ where $N,\tilde{N}$ are classical number operators for the right- and left- moving modes but now restricted to the transverse polarisations.