Free boson conformal field theory

The free boson conformal field theory has action quadratic in a scalar $X$ and central charge one. Its oscillator partition function is expressed through the Dedekind eta function.

For a noncompact free boson with target volume $V$, the torus partition function is $Z=V[2\pi \sqrt{\mathrm{Im}\tau }{]}^{-1}|\eta (\tau ){|}^{-2}$ in the normalization where the zero-mode Hamiltonian is $p^2/2$.

The Dedekind eta function isIt packages the oscillator contribution of a chiral free 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 $R\leftrightarrow {\alpha }^{\prime }/R$.