Lemma 3.6 . [042F]

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Lemma 3.6.

We have the identity

\begin{split}&\tilde{\alpha}_{1}(\mu_{1},\mu_{2},\eta)+\tilde{\alpha}_{1}(-\mu_{1},-\mu_{2},-\eta)\\ =&\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+\sum_{n\in\mathbb{Z}\setminus\{0\}}\{\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta+n|^{2}}}-\frac{1}{2\sqrt{a_{22}}|n|}\}\end{split}

where the RHS is recognized as the main part of the complex 2-dimensional Ooguri-Vafa potential. Similarly with $\mathfrak{D}_{i}$ for $i=1,2,3$ .