= Compact boson
{wiki}
A compact <boson> is a <scalar field> whose values lie on a circle, $X\sim X+2\pi R$. For a closed spatial <worldsheet> circle, a configuration may return to its initial value only after winding around the target circle: $X(\sigma+2\pi)=X(\sigma)+2\pi wR$, with integer $w$.
Single-valued target wavefunctions also quantize the centre-of-mass <momentum> to $n/R$. The integer $w$ is a winding number, whereas the corresponding <energy> scale is proportional to $|w|R/\alpha'$ in the string normalization. These are independent discrete labels, in addition to <string oscillator> excitations. Their opposite dependence on $R$ underlies <T-duality>: shrinking the circle makes <momentum> excitations heavier while winding excitations become lighter.
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