Poisson Resummation in CFT Bosonization

I am recently studying bosonization. In "the yellow book" by Philippe Di Francesco, Pierre Mathieu, and David Sénéchal, they calculated the partition function of a compactified boson using the path integral method. For each winding configuration $\phi(z+1,\bar z+1)=\phi(z,\bar z)+2\pi nR$ and $\phi(z+\tau,\bar z+\bar\tau)=\phi(z,\bar z)+2\pi wR$ (where we identified the torus as $\mathbb{T}^2\cong \mathbb{C}/\mathbb{Z}+\tau\mathbb{Z}$ ), the partition function is $\mathcal{Z}_{n,w}=\mathcal{Z}_{bos}\exp\{-\frac{\pi R^2|n\tau-w|^2}{2\Im(\tau)}\}$ where $\mathcal{Z}_{bos}$ is the partition function with periodic boundary condition. The total partition function is then $\mathcal{Z}=\sum_{n,w}\mathcal{Z}_{bos}\exp\{-\frac{\pi R^2|n\tau-w|^2}{2\Im(\tau)}\}$ After Poisson resummation, we get $\mathcal{Z}=\frac{1}{|\eta(\tau)|^2}\sum_{n,r}\frac{1}{2}q^{(r/R+nR/2)^2}\bar q^{(r/R-nR/2)^2}$ Alternatively, we can use canonical quantization to get the same result. In the canonical quantization language, $r$ is the label of $\Pi_0$ , the zero mode of $\Pi$ . I'm having trouble understanding why Poisson resummation over temporal windings gives us the sum over possible $\Pi_0$ eigenvalues. Mathematically it makes sense. But what does this Poisson resummation physically mean? Poisson resummation, if I understand correctly, is a Fourier transformation. I don't understand why after Fourier transformation, we get $\Pi_0$ eigenvalue labels.