Residual Gauge Transformations

In light-cone quantisation, we used world-sheet re-parametrisation invariance and Weyl invariance to set $h_{ab}={\eta }_{ab}$. However, this is only a partial gauge fixing, because there exists combinations of re-parametrisations and Weyl transformations, leaving this choice invariant. What are these residual gauge transformations?

The infinitesimal re-parametrisations $\delta {\sigma }^a+{\xi }^a$ of the world-sheet coordinates transform the world-sheet metric by

$\delta h^{ab}={\xi }^c{\partial }_c{h}^{ab}-h^{ac}{\partial }_c{\xi }^b-h^{bc}{\partial }_c{\xi }^a$.

Thus, setting $h_{ab}={\eta }_{ab}$, we find $\delta {\eta }^{ab}=-({\partial }^a{\xi }^b+{\partial }^b{\xi }^a)$. This transformation must be compensated by an infinitesimal Weyl transformation $\delta {\eta }^{ab}=\Theta {\eta }^{ab}$. Thus, these residual gauge transformations are solutions to

${\partial }^a{\xi }^b+{\partial }^b{\xi }^a=\Theta {\eta }^{ab}$.

In the light-cone coordinates,

${\partial }_{+}{\xi }^{-}={\partial }_{-}{\xi }^{+}=0$

${\partial }_{+}{\xi }^{+}+{\partial }_{-}{\xi }^{-}=\Theta$.

The general solution is given by arbitrary re-parametrisations of the form

${\xi }^{+}={\xi }^{+}({\sigma }^{+}){\xi }^{-}={\xi }^{-}({\sigma }^{-})$

combined with a Weyl transformation by $\Theta ={\partial }_{+}{\xi }^{+}+{\partial }_{-}{\xi }^{-}$.

Exponentiating, finite-residual gauge transformations correspond to re-parametrisations of the form

${\sigma }^{+}\to {\tilde{\sigma }}^{+}=f^{+}({\sigma }^{+}){\sigma }^{-}\to {\tilde{\sigma }}^{-}=f^{-}({\sigma }^{-})$.

In terms of the original coordinates $\tau$ and $\sigma$, this is

$\tau \to \tilde{\tau }=\frac{1}{2}(f^{+}(\tau +\sigma )+f^{-}(\tau -\sigma ))$

$\sigma \to \tilde{\sigma }=\frac{1}{2}(f^{+}(\tau +\sigma )-f^{-}(\tau -\sigma ))$.

The first equation means that by a residual gauge transformation, we may transform $\tau$ to any solution of the two-dimensional wave equation.

We can consider $X^{\mu }(\tau ,\sigma )$ (solutions of the two-dimensional wave equation) and $\tau$ can be transformed to be proportional to any one component of $X^{\mu }$. By doing this in a non-covariant manner, we can solve the Virasoro constraints.