In the paper"Level N Teichmüller TQFT and Complex Chern-Simons Theory" arXiv:1612.06986, the authors study the quantum dilogarithm function: \begin{equation} \mathrm{D}_{\rm b}(x,n)=\prod_{j=0}^{\infty}\frac{1-\mathfrak{q}^{2j+1}\exp\left[\frac{2\pi {\rm b}}{\sqrt{N}}x+\frac{2\pi i}{N}n\right]}{1-\widetilde{\mathfrak{q}}^{-2j-1}\exp\left[\frac{2\pi {\rm b}^{-1}}{\sqrt{N}}x-\frac{2\pi i}{N}n\right]},\quad x\in\mathbb{C},\quad n\in\mathbb{Z}/N\mathbb{Z} \end{equation} where $\mathfrak{q}=e^{\frac{\pi i}{N}({\rm b}^2+1)}$ and $\widetilde{\mathfrak{q}}=e^{\frac{\pi i}{N}({\rm b}^{-2}+1)}$ with odd integer $N$ and ${\rm b}$ satisfying $|{\rm b}|=1$, $\mathrm{Re}({\rm b})>0$, $\mathrm{Im}({\rm b})>0$. The authors state the following theorem without proof:
Theorem: Suppose $\operatorname{Im}(\mathrm{b})>0$ and $N$ odd, and let $u, v, w \in \mathbb{C}$ and $a, b, c \in \mathbb{Z} / N \mathbb{Z}$ satisfy $$ \operatorname{Im}\left(v+\frac{c_{\mathrm{b}}}{\sqrt{N}}\right)>0, \quad \operatorname{Im}\left(-u+\frac{c_{\mathrm{b}}}{\sqrt{N}}\right)>0, \quad \operatorname{Im}(v-u)<\operatorname{Im}(w)<0. $$ where $c_{\rm b}=\frac{i}{2}({\rm b}+{\rm b}^{-1})$.
Define
$$ \Psi(u, v, w, a, b, c) \equiv \frac{1}{\sqrt{N}}\sum_{d\in\mathbb{Z}/N\mathbb{Z}}\int_{\mathbb{R}}\mathrm{d}x\, \frac{\mathrm{D}_{\mathrm{b}}(x+u, a+d)}{\mathrm{D}_{\mathrm{b}}(x+v, b+d)} e^{2 \pi i w x} e^{-2 \pi i \frac{c d}{N}}. $$
Then we have that
$$ \begin{aligned} & \Psi(u, v, w, a, b, c) \\ & =\zeta_0 \frac{\mathrm{D}_{\mathrm{b}}\left(v-u-w+\frac{c_{\mathrm{b}}}{\sqrt{N}}, b-a-c\right)}{\mathrm{D}_{\mathrm{b}}\left(-w-\frac{c_{\mathrm{b}}}{\sqrt{N}},-c\right) \mathrm{D}_{\mathrm{b}}\left(v-u+\frac{c_{\mathrm{b}}}{\sqrt{N}}, b-a\right)} e^{2 \pi i w\left(\frac{c_{\mathrm{b}}}{\sqrt{N}}-u\right)} \omega^{a c} \\ & =\zeta_0^{-1} \frac{\mathrm{D}_{\mathrm{b}}\left(w+\frac{c_{\mathrm{b}}}{\sqrt{N}}, c\right) \mathrm{D}_{\mathrm{b}}\left(-v+u-\frac{c_{\mathrm{b}}}{\sqrt{N}},-b+a\right)}{\mathrm{D}_{\mathrm{b}}\left(-v+u+w-\frac{c_{\mathrm{b}}}{\sqrt{N}},-b+a+c\right)} e^{2 \pi i w\left(-\frac{c_{\mathrm{b}}}{\sqrt{N}}-v\right)} \omega^{b c} \end{aligned} $$ where $\zeta_0=e^{-\pi i\left(N-4 c_b^2 N^{-1}\right) / 12}$ and $\omega=e^{2\pi i/N}$.
Could anyone give some details of the proof?
