I'm trying to understand the existence of the light-cone gauge for the closed bosonic string in a more mathematical precise language.
Namely, when looking for critical points of Polyakov action, we can
Choose local coordinates (in case of cylinder, these are called $(\sigma,\tau)$, more generally let's call them $(\sigma_1 , \sigma_2$))
Write the world sheet metric $h = \exp(2f)\eta$ over conceivable a new, possibly smaller coordinate patch (abusing notation, I think physics texts are OK if we just call the restricted coordinates of the new patch $(\sigma_1,\sigma_2)$ again)
Ignore the exponential prefactor by Weyl invariance.
At this point, the E.o.M are free wave equations for each coordinate of the string in addition to the two Virasoro constraints.
- Define "light-cone" coordinates $\sigma^{\pm} = \sigma_1 \pm \sigma_2$. At this point, one solves easily the wave equations by Fourier expansions and can see the $M^2 = \sum (\mathrm{Fourier \ modes})$ formula.
Yet somehow physics textbooks say there is still gauge redundancy remaining on the worldsheet.
With a mathematical background in differential geometry, I am having trouble understanding what physics texts are expressing here. We have a specific local chart on the world sheet and spacetime too. Why is there still redundancy? And how do we know the light-cone gauge exists?
