Compact boson

A compact boson takes values on a circle, $X\sim X+2\pi R$. Its states carry quantized momentum $n/R$ and integer winding $wR$.

For a compact boson of radius $R$, momentum number $n$ and winding number $w$ combine into left- and right-moving zero-mode momenta $p_L=n/R+wR$ and $p_R=n/R-wR$ in a common normalization.

T-duality identifies a compact boson of radius $R$ with one of reciprocal radius by exchanging momentum and winding numbers. Factors of ${\alpha }^{\prime }$ restore the dimensional relation $R\leftrightarrow {\alpha }^{\prime }/R$.