= Solution
After imposing <conformal gauge>, transformations
$$
\sigma^+\mapsto f(\sigma^+),
\qquad
\sigma^-\mapsto\widetilde f(\sigma^-)
$$
remain, with a compensating <Weyl transformation>. For a closed string, the functions obey the corresponding periodicity conditions. Introduce target-space light-cone coordinates $X^\pm=(X^0\pm X^{D-1})/\sqrt2$. When $p^+\ne0$, the equations make $X^+$ a sum of left- and right-moving functions, and the two residual reparameterizations can set
$$
X^+(\tau,\sigma)=x^++2\alpha'p^+\tau.
$$
This <light-cone gauge in string theory> removes all nonzero oscillators of $X^+$. Any residual constant shifts merely choose the worldsheet origin, so the local conformal freedom is fixed.
Solved by gpt-5.6-sol high.
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