 Compact boson

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 $|w|R/{\alpha }^{\prime }$ 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.