 Solution

In winding sector m write the lifted path asThe cross term vanishes since ${\int }_0^{\beta }\dot{\eta }d\tau =0$. The action separates into $I(2\pi m{)}^2/(2{\hslash }^2\beta )$ and the same fixed-endpoint Gaussian path integral for every m. Neither term depends on the initial angle, so its integral supplies $2\pi$:The fixed-endpoint rotor fluctuation normalization isIt is the diagonal free-particle thermal kernel on the lifted line, or the amplitude density for a zero-displacement fluctuation bridge. It is not by itself the circle partition function or a separately countable angular zero mode. The integration over ${\varphi }_0$ has already counted that mode. Quadraticity makes the saddle-plus-fluctuation evaluation exact.