Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 1 / c / Solution

For a target circle of radius $R$, maps may wind, so the boundary condition isSingle-valued target-space wavefunctions quantize the center-of-mass momentum as $p_Y=n/R$, with $n\in \mathbb{Z}$. The compact boson expansion becomesEquivalently, its left- and right-moving zero-mode momenta areThe integers $n$ and $w$ are the momentum and winding modes.