= Solution
For a target circle of radius $R$, maps may wind, so the boundary condition is
$$
Y(\tau,\sigma+2\pi)=Y(\tau,\sigma)+2\pi Rw,
\qquad w\in\mathbb Z.
$$
Single-valued target-space wavefunctions quantize the center-of-mass momentum as $p_Y=n/R$, with $n\in\mathbb Z$. The <compact boson> expansion becomes
$$
Y=y+\alpha'\frac nR\tau+wR\sigma
+i\sqrt{\frac{\alpha'}2}\sum_{r\ne0}\frac1r
\left(\alpha_r e^{-ir(\tau-\sigma)}
+\widetilde\alpha_r e^{-ir(\tau+\sigma)}\right).
$$
Equivalently, its left- and right-moving zero-mode momenta are
$$
p_L=\frac nR+\frac{wR}{\alpha'},
\qquad
p_R=\frac nR-\frac{wR}{\alpha'}.
$$
The integers $n$ and $w$ are the <momentum and winding modes>.
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