I have a problem understanding a part of https://arxiv.org/pdf/hep-th/0005247.pdf on page 59.
Here, $$ \widetilde{W}=-N\sigma(\log\sigma-1)-t\sigma. \label{SlogS} $$ is a function of the complex parameter $\sigma$. $N$ is an integer, while $t$ is a complex parameter. Using the notation $$ t_{\it eff}=t+N\log\sigma, $$ and $t_{\it eff}-N=e^{i\gamma_{\it eff}}|t_{\it eff}-N|$, we have $$ \widetilde{W}=-e^{i\gamma_{eff}(\sigma)}\sigma|t_{eff}(\sigma)-N| \\=-\bigg[\textrm{Re }(e^{i\gamma_{eff}(\sigma)}\sigma)+i\textrm{Im }(e^{i\gamma_{eff}(\sigma)}\sigma)\bigg]|t_{eff}(\sigma)-N|\tag{1} $$ The authors claim that the solution for $\sigma$ such that we have a straight line in the $\widetilde{W}$-plane is $${\rm Im}\Bigl(e^{i\gamma_{\it eff}(\sigma)}\sigma\Bigr)= {\rm constant},$$ how does one show this to be true?
EDIT
There is additional information which I did not include previously, which is consistent with what Andrew D. Hwang deduced. There is a physical constraint (B-type supersymmetry) which insists that the straight line is parallel to the real line, i.e., it is horizontal. In other words, we should have $\textrm{Im} (\widetilde{W})=\textrm{constant}$. This was discussed earlier in the paper (above equation 3.13). However, from $(1)$ it seems to me that this requires $${\rm Im}\Bigl(e^{i\gamma_{\it eff}(\sigma)}\sigma\Bigr)|t_{eff}(\sigma)-N|= {\rm constant},$$ and not what is given above.