Torus partition function of a free boson

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.