= Solution
Choose a real lift $\Delta\theta=\theta_f-\theta_i$. Classical paths fall into <winding number> sectors $j\in\mathbb Z$ and are
$$
\theta_j(t)=\theta_i+\frac{\Delta\theta+2\pi j}{T}t,
\qquad
S_j=\frac{mR^2}{2T}(\Delta\theta+2\pi j)^2.
$$
Each sector has the same fluctuation determinant, so the image-sum form of the propagator is
$$
K(\theta_f,T;\theta_i,0)
=\sqrt{\frac{mR^2}{2\pi i\hbar T}}
\sum_{j\in\mathbb Z}e^{iS_j/\hbar}.
$$
Set $a=\hbar T/(2mR^2)$. Applying the <Poisson summation formula> to the Gaussian $e^{ix^2/(4a)}$ gives
$$
\frac1{\sqrt{4\pi ia}}
\sum_{j\in\mathbb Z}e^{i(\Delta\theta+2\pi j)^2/(4a)}
=\frac1{2\pi}\sum_{n\in\mathbb Z}e^{in\Delta\theta-ian^2},
$$
which is exactly the spectral propagator found in part i. Thus the angular-momentum sum is dual to a sum over homotopy classes of classical paths.
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