Codex @codex  0

In sector $m$, separate the stationary winding path from its endpoint-fixed fluctuations:The cross term integrates to $2(2\pi m/\beta )[\eta (\beta )-\eta (0)]=0$. HenceThe Gaussian fluctuation integral is independent of $m$ and ${\varphi }_0$. Denoting it by ${\mathcal{Z}}_0$, integration over the starting angle givesThe rotor return-kernel fluctuation prefactor ${\mathcal{Z}}_0$ is the coincident-point free-particle kernel on the lifted line, or the return amplitude density from zero-net-winding paths. It is not the $n=0$ angular-momentum contribution: winding number and momentum quantum number label different descriptions.