= Solution
The <Euler-Lagrange equation> of the gauge-fixed <Polyakov action> is the two-dimensional wave equation
$$
(\partial_\tau^2-\partial_\sigma^2)X^\mu=0.
$$
Its general solution is a sum of left- and right-moving functions. Periodicity in $\sigma$ and a Fourier expansion give the <closed-string mode expansion>
$$
X^\mu=x^\mu+\alpha'p^\mu\tau
+i\sqrt{\frac{\alpha'}2}\sum_{r\ne0}\frac1r
\left(\alpha_r^\mu e^{-ir(\tau-\sigma)}
+\widetilde\alpha_r^\mu e^{-ir(\tau+\sigma)}\right).
$$
The zero modes satisfy $\alpha_0^\mu=\widetilde\alpha_0^\mu=\sqrt{\alpha'/2},p^\mu$. Here $x^\mu$ and $p^\mu$ are the center-of-mass position and momentum, while the nonzero <string oscillators> describe shape excitations.
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