Momentum-winding duality of a quantum rotor (${\sum }_n{e}^{-\beta {\hslash }^2(n-f{)}^2/(2I)}=\sqrt{2\pi I/(\beta {\hslash }^2)}{\sum }_m{e}^{-2{\pi }^{2}I{m}^2/(\beta {\hslash }^2)+2\pi ifm}$)

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 $\sqrt{I/(2\pi {\hslash }^2\beta )}$; 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.