= Momentum-winding duality of a quantum rotor
{title2=$\sum_n e^{-\beta\hbar^2(n-f)^2/(2I)}=\sqrt{2\pi I/(\beta\hbar^2)}\sum_m e^{-2\pi^2Im^2/(\beta\hbar^2)+2\pi ifm}$}
The thermal <trace> can be expanded in angular-momentum eigenstates or in lifted paths whose endpoints differ by $2\pi m$. A winding saddle is linear in imaginary time, and its flux weight is the <Aharonov-Bohm phase> $e^{2\pi ifm}$. The common fixed-endpoint fluctuation factor is $\sqrt{I/(2\pi\hbar^2\beta)}$; integrating the initial angle supplies $2\pi$. The <Poisson summation formula> converts the resulting winding Gaussian sum into the <momentum> sum with exactly cancelling prefactors. The Gaussian evaluation is exact for this free rotor.
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