The periods along the generating cycles in the $T^{2}$-fibres are straightforward:

and $\int_{S^{1}}\tilde{\zeta}_{0}^{\prime}=-\sqrt{-1}\int_{S^{1}}\vartheta_{1}+\vartheta_{2}=-4\pi\sqrt{-1}$.

Computing the period along the other $S^{1}$ requires a special trick. As a preparatory subtle remark, the Kähler metric is not globally defined over the base $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ due to incompleteness issues, but the quantities $\tilde{v}^{ij},\tilde{w},\vartheta_{i}$ make sense globally. Consider the $S^{1}$ on the base defined by $\{\mu_{1}=\mu_{2}=0,y=\text{const}\}$. If we attempt to lift this $S^{1}$ by parallel transport, in general we cannot get a closed loop, and this failure is measured by the holonomy of the $T^{2}$-connection $\vartheta=(\vartheta_{1},\vartheta_{2})$ along the $S^{1}$. When $y\to+\infty$, due to the exponential decay of the $x$-dependent part of $\tilde{v}^{ij},\tilde{w},\vartheta_{i}$, this holonomy converges to two real numbers $(\theta_{1}^{\infty},\theta_{2}^{\infty})$ modulo $2\pi\mathbb{Z}$. In particular, if we twist $\vartheta$ by a flat $T^{2}$-connection, then $\theta_{1}^{\infty},\theta_{2}^{\infty}$ receive a corresponding twist so that $\tilde{\zeta}_{i}$ is unaffected. Thus we can assume without loss of generality that $\theta_{i}^{\infty}=0$, namely the asymptotic holonomy of $\vartheta$ is zero, so in the limit the $S^{1}$ cycle lifts to a closed loop, on which we can evaluate the period asymptotically.

By construction $\int_{S^{1}}\vartheta_{i}=0$, and using $\tilde{\beta}_{i}(0,0,\eta)=\beta_{i}(0,0,1)\pi\cot(\pi\eta)$ from the proof of Lemma 3.7, we compute

From this we see the integrality condition on the periods, so the holomorphic functions $Z_{i}$ are well defined without multivalue issues.

Notice the definition of $Z_{i}$ for $i=0,1,2$ involve three unspecified multiplicative constants; by prescribing their product appropriately, the functional equation follows from Lemma 3.9. The remaining two free multiplicative constants will be fixed in later Sections. ∎