&\quad -\frac{1}{(2\pi)^3}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\left(\partial_{\tau_2}\partial_{\overline{z}_{\hat{\gamma}}}\partial_{\tau_2}\mathcal{I}_{\hat{\gamma}}^{(3)}-\frac{1}{\tau_2}\partial_{\overline{z}_{\hat{\gamma}}}\partial_{\tau_2}\mathcal{I}_{\hat{\gamma}}^{(3)}\right)\D \tau_2-\frac{\I \tau_2\chi}{(4\pi)^2}\partial_{\zeta^0}\partial_{\tau_2}\mathcal{I}_{0}^{(2)}\D \tau_2\,,\\
\end{split}
\end{equation}

so using again Lemma <ref> one finds that

\begin{equation}\label{typeiibthetaplusinst2}
\begin{split}
&2R^2\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi \I n\zeta_{\gamma}}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D \widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}+\frac{\D \overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}+\frac{2}{\tau_2}\D \tau_2\right)\\
&=\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}e^{-S_{\hat{\gamma},m,n}}\Bigg[\frac{\tau_2^3}{8\pi}q_a\frac{m}{|m\tau+ n|^2}\left(\frac{1}{|m\tau+ n|}+2\pi q_bt^b\right)\D z^a\\
&\quad +\frac{\tau_2}{8\pi}q_a\frac{m\tau_1+ n+|m\tau+ n|}{m|m\tau+ n|^2} \left(\frac{|m\tau+ n|-(m\tau_1+ n)}{|m\tau+ n|}-2\pi q_at^a(m\tau_1+ n+|m\tau+ n|)\right)\D \overline{z}^a\\
&\quad+\frac{\I  \tau_2^2}{4\pi}\left(\frac{2\pi q_at^a}{|m\tau+ n|^2} +\frac{1}{|m\tau+ n|^3}\right)(m\tau_1+ n+|m\tau+ n|)\frac{mq_bt^b}{|m\tau+ n|}\D \tau_2\\
&\quad+\frac{\I  \tau_2^2}{8\pi^2}m\left(\frac{4\pi q_at^a}{|m\tau+ n|^4} +\frac{3}{|m\tau+ n|^5}\right)(m\tau_1+ n+|m\tau+ n|)\D \tau_2-\frac{\I  \tau_2^2}{8\pi^2}m\left(\frac{4\pi q_bt^b}{|m\tau+ n|^3}+\frac{3}{|m\tau+ n|^4}\right)\D \tau_2\Bigg]\\
&\quad-\frac{3\I \tau_2^2\chi}{(4\pi)^2}\sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}\frac{m(m\tau_1+ n)}{|m\tau+ n|^5}\D \tau_2\,.
\end{split}
\end{equation}

On the other hand, using the definitions of $\sigma^{\text{inst}}$, $\widetilde{\zeta}_i^{\text{inst}}$ and $\mathfrak{F}^{\text{w.s.}}$ in (<ref>) and (<ref>), one can compute $\theta_{+}^{P,\text{w.s.}}$. Summing the contributions of $\theta_{+}^{P,\text{w.s.}}$ to (<ref>) and (<ref>), and multiplying the whole result by $\I$, one then obtains (<ref>).

§.§.§ Proof of Theorem <ref>

Now we have all the preliminary results needed to prove Theorem <ref>:

First, we remark that the functions $(\xi^i,\widetilde{\xi}_i^{\text{cl}},\alpha^{\text{cl}})$ satisfy

\begin{equation}\label{dccl}
-2\pi \I(\D \alpha^{\text{cl}}+\widetilde{\xi}_i^{\text{cl}}\D \xi^i-\xi^i\D \widetilde{\xi}_i^{\text{cl}})=f^{\text{cl}}\frac{\D t}{t}+t^{-1}\I\theta_+^{P,\text{cl}} -2\I\theta_3^{P,\text{cl}} -t\I\theta_-^{P,\text{cl}}\,.
\end{equation}

The proof of (<ref>) is almost the same as Proposition <ref>, but we replace in the proof $t\to -\I t$ (recall that we did this rescaling at the beginning of Section <ref>) and drop the $\log(t)$ term of $\alpha$ in (<ref>),  while keeping in mind that $f^{\text{cl}}$ differs from $f$ in Proposition <ref> by $16\pi c_\ell$, compare (<ref>), where $f^{\text{w.s}}$ and $f^{\text{inst}}$ need to be set to zero to get
the $f$ in Proposition <ref>. We then obtain immediately the following identity

\begin{equation}
4\pi \I \left(-\frac{1}{2}\D(\alpha^{\text{cl}}-\xi^i\widetilde{\xi}_i^{\text{cl}}) - \widetilde{\xi}_i^{\text{cl}}\D \xi^i \right)=f^{\text{cl}}\frac{\D t}{t}+t^{-1}\I\theta_+^{P,\text{cl}} -2\I\theta_3^{P,\text{cl}} -t\I\theta_-^{P,\text{cl}}\,.
\end{equation}

In particular, to prove Theorem <ref> it is enough to show using the decompositions at the beginning of Section <ref> that

\begin{equation}\label{contactsimplication}
4\pi\I\left(\D \alpha^{\text{inst}} -\widetilde{\xi}_i^{\text{inst}}\D \xi^i\right)=(-16\pi c_{\ell}+f^{\text{w.s.}}+f^{\text{inst}})\frac{\D t}{t}+t^{-1}\I (\theta_+^{P,\text{w.s.}}+\theta_+^{P,\text{inst}}) -2\I (\theta_3^{P,\text{w.s.}}+\theta_3^{P,\text{inst}}) -t\I \overline{(\theta_+^{P,\text{w.s.}}+\theta_+^{P,\text{inst}})},
\end{equation}

where we have defined

\begin{equation}\label{dcdiff}
\alpha^{\text{inst}}:=\alpha+\frac{1}{2}(\alpha^{\text{cl}}-\xi^{i}\widetilde{\xi}_i^{\text{cl}}), \quad \widetilde{\xi}_i^{\text{inst}}:=\widetilde{\xi}_i-\widetilde{\xi}_i^{\text{cl}}\,.
\end{equation}

In order to do this, one can use the explicit formulas (<ref>) to compute the left-hand side of (<ref>). In terms of the coordinates $(\tau_2,b^a,t^a,\zeta^i,\widetilde{\zeta}_i,\sigma)$, the left-hand side has  only $\D t$, $\D t^a$, $\D b^a$, $\D \zeta^i$ and $\D \tau_2$ components, since $\alpha^{\text{inst}}$ and $\xi^i$ do not depend on $\widetilde{\zeta}_i$ and $\sigma$. Recall that $R=\tau_2/2$. For the $\mathrm{d}t$ component, one obtains

\begin{equation}\label{dtcompcomputation}
\begin{split}
4\pi \I&\left(\D \alpha^{\text{inst}}-\widetilde{\xi}_i^{\text{inst}}\D \xi^i\right)\Big|_{\D t}\\
=&\frac{\tau_2^2}{2(2\pi)^2}\sum_{\hat{\gamma}\in \Lambda^{+}\cup\{0\}}n_{\hat{\gamma}}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}(m\tau_1+n)(t^{-2}+1)\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^4}\\
&+\frac{\tau_2^2}{2(2\pi)^2}\sum_{\hat{\gamma}\in \Lambda^{+}\cup\{0\}}n_{\hat{\gamma}}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left((m\tau_1+n)(t^{-1}-t) -2m\tau_2\right)\frac{1+(t_{+}^{m,n})^2}{(t-t_{+}^{m, n})^2}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^4}\\
&-\frac{\tau_2^2}{2(2\pi)^2}(t^{-2}+1)\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(\frac{1}{m\xi^0+n}+\frac{m\tau_1+n}{|m\tau+ n|^2}\right)\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau + n|^2}\\
&+\frac{\tau_2^2}{2\pi}t^{-1}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}q_at^a\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^2}\\
=&\frac{\tau_2^2}{2(2\pi)^2}\sum_{\Lambda^{+}\cup\{0\}}n_{\hat{\gamma}}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left[\left(\frac{(m\tau_1+n)(t^{-1}-t)-2m\tau_2}{|m\tau+n|^2}\right)\frac{1+(t_{+}^{m,n})^2}{(t-t_{+}^{m, n})^2}-\frac{(t^{-2}+1)}{m\xi^0+n}\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}\right]\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^2}\\
&+\frac{\tau_2^2}{2\pi}t^{-1}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}q_at^a\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^2}\,,
\end{split}
\end{equation}

where in the first equality we are using in the case $m=0$ equation (<ref>) and

\begin{equation}\label{m=0def2}
\frac{1+(t_{+}^{0, n})^2}{(t-t_{+}^{0, n})^2}:=\begin{cases}
-\frac{\D}{\D t}\left(\frac{1+t_{+}^{0, n}t}{t-t_{+}^{0, n}}\right)=1/t^2, \quad n<0\\
-\frac{\D}{\D t}\left(\frac{1+t_{+}^{0, n}t}{t-t_{+}^{0, n}}\right)=1, \quad n>0
\end{cases},
\end{equation}
while in last equation we left the last sum the same and grouped the rest of the terms (notice that two sums cancel each other). For the term in square brackets in (<ref>), we use for $m\neq 0$ that $t\cdot (m\xi^0+n)=-mR(t-t_{+}^{m,n})(t-t_{-}^{m,n})$ and $t_+^{m,n}t_{-}^{m,n}=-1$, together with the fact that $t_+^{m,n}+t_{-}^{m,n}=2(m\tau_1+n)/m\tau_2$ and $t_+^{m,n}-t_{-}^{m,n}=2|m\tau+n|/m\tau_2$, see (<ref>). We then obtain the following for the case $m\neq 0$:

\begin{align*}
&\left(\frac{(m\tau_1+n)(t^{-1}-t)-2m\tau_2}{|m\tau+n|^2}\right)\frac{1+(t_{+}^{m,n})^2}{(t-t_{+}^{m, n})^2}-\frac{(t^{-2}+1)}{m\xi^0+n}\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}\\
&=\frac{4}{m\tau_2 t(t_{+}^{m,n}-t_{-}^{m,n})}=\frac{2}{t|m\tau+n|}\,.\numberthis
\end{align*}

In the case $m=0$ we use (<ref>) and (<ref>), so that

\begin{equation}
\left[\left(\frac{(m\tau_1+n)(t^{-1}-t)-2m\tau_2}{|m\tau+n|^2}\right)\frac{1+(t_{+}^{m,n})^2}{(t-t_{+}^{m, n})^2}-\frac{(t^{-2}+1)}{m\xi^0+n}\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}\right]\Big|_{m=0}=\frac{2}{t|n|}\,.
\end{equation}

Joining everything together we find

\begin{equation}
4\pi \I\left(\D \alpha^{\text{inst}}-\widetilde{\xi}_i^{\text{inst}}\D \xi^i\right)\Big|_{\D t}=t^{-1}\frac{\tau_2^2}{(2\pi)^2}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(\frac{1}{|m\tau+ n|}+2\pi q_at^a\right)\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau + n|^2}\,.
\end{equation}

In particular, we find using Proposition <ref> that

\begin{equation}
4\pi \I \left(\D \alpha^{\text{inst}}-\widetilde{\xi}_i^{\text{inst}}\D \xi^i\right)\Big|_{\D t}=t^{-1}\left(f^{\text{w.s.}}+f^{\text{inst}}-\frac{\chi}{12}\right)\;,
\end{equation}

matching the required term on the right-hand side of (<ref>) provided that $c_{\ell}=\frac{\chi}{192\pi}$. On the other hand, for the other components $\D t^a$, $\D b^a$, $\D \zeta^i$ and $\D \tau_2$ of the left-hand side of (<ref>), one needs to show that they decompose into three summands with factors $t^{-1}$, $t^0$ and $t$, which should match the corresponding component of $\I (\theta_+^{P,\text{w.s.}}+\theta_+^{P,\text{inst}})$, $-2\I (\theta_3^{P,\text{w.s.}}+\theta_3^{P,\text{inst}})$, and $-\I \overline{(\theta_+^{P,\text{w.s.}}+\theta_+^{P,\text{inst}})}$ in (<ref>), respectively. In particular, the summand with the $t$ factor should correspond to the conjugate of the $t^{-1}$ summand, for each component. For example the computation for the $\D t^a$ component gives

\begin{align*}
&4\pi\I\left(\D \alpha^{\text{inst}}-\widetilde{\xi}_i^{\text{inst}}\D \xi^i\right)\Big|_{\D t^a}\\
&=\frac{\tau_2^2}{4\pi}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}q_a\sum_{(m,n)\neq (0,0)} \left(\frac{m\tau_1+ n}{|m\tau+ n|}(t^{-1}-t)-\frac{2m\tau_2}{|m\tau+ n|}+(t^{-1}+t)\right)\frac{1+t_+^{m, n}t}{t-t_+^{m, n}}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^2}\\
&=\frac{\tau_2^2}{4\pi}\sum_{\hat{\gamma}\in \Lambda^{+}\cup\{0\}}n_{\hat{\gamma}}q_a\sum_{(m,n)\neq (0,0)} \left(-t^{-1}\frac{m\tau_2}{|m\tau+ n|}-2\frac{m\tau_1+ n}{|m\tau+ n|} +t\frac{m\tau_2}{|m\tau+ n|}\right)\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^2}\\ \numberthis
\end{align*}

where for the last equality we used the identities for $t_{\pm}^{m,n}$ used in the computation of the $\D t$ component. Comparing with Proposition <ref> and <ref> one readily sees that the $t^{-1}$ and $t^{0}$ summands match the corresponding $\D t^a$ component. Furthermore, the summand with the $t$-factor is seen to correspond to the conjugate of the summand with the $t^{-1}$ factor, since $\overline{S_{\hat{\gamma},m,n}}=S_{\hat{\gamma},-m,-n}$, so the summand with the $t$-factor also gives the required contribution.

By a similar (albeit tedious) computation one can check using Proposition <ref> and <ref> that the remaining components on the left-hand side and right-hand side of  (<ref>) match, showing that (<ref>) indeed define Darboux coordinates for the contact structure associated to instanton corrected q-map spaces.

§.§ S-duality

Recall that we have the S-duality action on $\overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}$, defined by (<ref>). We start by defining a lift of the S-duality action from $\overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}$ to $\overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}\times \mathbb{C}P^1$, following [15, 11].

Given an element

\begin{equation}
a & b \\
c & d \\
\end{pmatrix}\in \mathrm{SL}(2,\mathbb{Z})
\end{equation}
we lift the action of $A$ from $\overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}$ to  $\overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}\times \mathbb{C}P^1$ by defining the action on the fiber coordinate $t\in \mathbb{C}P^1$ over $(\tau_1+\I\tau_2,b^a+\I t^a, c^a,c_a,c_0,\psi)\in \overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}$ by:

\begin{equation}\label{sdualitylift}
\begin{pmatrix}
a & b\\
c & d\\
\end{pmatrix}\cdot t:=\begin{dcases}
t \quad \text{if $c=0$ and $a>0$}\\
-1/t \quad \text{if $c=0$ and $a<0$}\\
\frac{1+t^{c,d}_{+}t}{t^{c,d}_{+}-t} \quad \text{if $c\neq 0$}
\end{dcases} \,.
\end{equation}

Since $\mathrm{SL}(2,\mathbb{Z})$ is generated by

\begin{equation}\label{sl2gen}
1 & 1\\
0 & 1\\
\end{pmatrix}, \quad S=\begin{pmatrix}
0 & -1\\
1 & 0\\
\end{pmatrix}
\end{equation}

it is not hard to check that (<ref>) defines a lift of the S-duality action. We note that the transformation on $t$ depends on the base point in the case where $c\neq 0$, since $t_{+}^{c,d}$ depends on $\tau$. We further remark that by using the fact that $t_{+}^{c,d}t_{-}^{c,d}=-1$ the $t$-variable transformation when $c\neq 0$ can be rewritten as

\begin{equation}
\begin{pmatrix}
a & b\\
c & d\\
\end{pmatrix}\cdot t=\frac{1+t^{c,d}_{+}t}{t^{c,d}_{+}-t}=-\frac{t_{-}^{c,d}-t}{1+t_{-}^{c,d}t}\,.
\end{equation}

Now let $(\widetilde{N},g_{\overline{N}})$ be an instanton corrected q-map space constructed in Section <ref>.  By using the mirror map (<ref>), we can identify $\widetilde{N}\subset \overline{\mathcal{N}}_{\text{IIA}}$ with an open subset of $\mathcal{M}^{-1}(\widetilde{N})\subset \overline{\mathcal{N}}_{\text{IIB}}\subset \overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}$. It then follows that its twistor space satisfies $\mathcal{Z}\cong \widetilde{N}\times \mathbb{C}P^1\subset \overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}\times \mathbb{C}P^1$. If $A\in \mathrm{SL}(2,\mathbb{Z})$ leaves $\widetilde{N}\subset \overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}$ invariant, then we get an induced action of $A$ on $\mathcal{Z}$. Assuming that $A$ acts on $\mathcal{Z}$, we now show the key transformation property of the Darboux coordinates, following again [15, 11]:
Let $(\widetilde{N},g_{\overline{N}})$ be an instanton corrected q-map space with $1$-loop parameter $c_{\ell}=\frac{\chi}{192\pi}$ such that $\widetilde{N}$ is invariant under

\begin{equation}
a & b \\
c & d \\
\end{pmatrix}\in \mathrm{SL}(2,\mathbb{Z})\,.
\end{equation}

Furthermore consider the Darboux coordinates $(\xi^i,\widetilde{\xi}_i,\widetilde{\alpha})$ of $\mathcal{Z}$, where  $\widetilde{\alpha}:=\alpha -\xi^i\widetilde{\xi}_i$ and $(\xi^i,\widetilde{\xi}_i,\alpha)$ are the Darboux coordinates given by (<ref>). Then $(\xi^i,\widetilde{\xi}_i,\widetilde{\alpha})$  satisfy the following transformation under the lift of the action of $A$ given in (<ref>):

\begin{equation}\label{twistorSduality}
\begin{split}
\xi^0&\to \frac{a\xi^0+b}{c\xi^0+d}, \quad \xi^a \to \frac{\xi^a}{c\xi^0+d}, \quad \widetilde{\xi}_a \to \widetilde{\xi}_a +\frac{c}{2(c\xi^0+d)}\kappa_{abc}\xi^b\xi^c\\
\begin{pmatrix}
\widetilde{\xi}_0 \\
\widetilde{\alpha}
\end{pmatrix}&\to \begin{pmatrix}
d & -c\\
-b & a \\
\end{pmatrix}\begin{pmatrix}
\widetilde{\xi}_0 \\
\widetilde{\alpha}
\end{pmatrix} +\frac{1}{6}\kappa_{abc}\xi^a\xi^b\xi^c\begin{pmatrix}
\end{pmatrix}.
\end{split}
\end{equation}

To show the required transformation rule we follow the same argument of [15], which we include here with more detail. It is straightforward to check that the classical coordinates $(\xi^i,\widetilde{\xi}_i^{\text{cl}},\widetilde{\alpha}^{\text{cl}})$, where $\widetilde{\alpha}^{\text{cl}}:=-\frac{1}{2}(\alpha^{\text{cl}}-\xi^i\widetilde{\xi}_i^{\text{cl}})-\xi^{i}\widetilde{\xi}_i^{\text{cl}}=-\frac{1}{2}(\alpha^{\text{cl}}+\xi^i\widetilde{\xi}_i^{\text{cl}})$ satisfy (<ref>) [13]. Hence, it is enough to show that $\widetilde{\xi}_i^{\text{inst}}=\widetilde{\xi}_i-\widetilde{\xi}_i^{\text{cl}}$ and $\widetilde{\alpha}^{\text{inst}}=\widetilde{\alpha}-\widetilde{\alpha}^{\text{cl}}$ transform under under the action of $A\in \mathrm{SL}(2,\mathbb{Z})$ by

\begin{equation}\label{insttransrules}
\widetilde{\xi}_a^{\text{inst}}\to \widetilde{\xi}_a^{\text{inst}}, \quad  \begin{pmatrix}
\widetilde{\xi}_0^{\text{inst}} \\
\widetilde{\alpha}^{\text{inst}}
\end{pmatrix}\to \begin{pmatrix}
d & -c\\
-b & a \\
\end{pmatrix}\begin{pmatrix}
\widetilde{\xi}_0^{\text{inst}} \\
\widetilde{\alpha}^{\text{inst}}
\end{pmatrix}.
\end{equation}

Using that (<ref>) and (<ref>), we find that  $\widetilde{\alpha}^{\text{inst}}=\alpha^{\text{inst}}-\xi^i\widetilde{\xi}_i^{\text{inst}}$ and

\begin{equation}
\begin{split}
\widetilde{\alpha}^{\text{inst}}&=-\frac{\I\tau_2}{2(2\pi)^3}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}^{(0)}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(\frac{\xi^0}{m\xi^0+ n}+\frac{m|\tau|^2+ n\tau_1}{|m\tau+ n|^2}\right)\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau + n|^2}\\
&+\frac{\tau_2}{2(2\pi)^2}\sum_{\hat{\gamma}\in \Lambda_{+}\cup \{0\}}n_{\hat{\gamma}}^{(0)} \sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(q_ac^a \frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}+\I q_at^a\left(\tau_1\frac{1-t_{+}^{m, n}t}{t-t_{+}^{m, n}}-\tau_2\frac{t+t_{+}^{m, n}}{t-t_{+}^{m, n}}\right)\right)\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^2}\,.
\end{split}
\end{equation}

where, in addition to (<ref>), we have defined

\begin{equation}
\frac{t+t_{+}^{0,n}}{t-t_{+}^{0,n}}:=\begin{dcases}
-1 \quad n>0\\
1 \quad n<0\\
\end{dcases}\,.
\end{equation}

We start by showing that $\widetilde{\xi}_a^{\text{inst}}$ is invariant under the S-duality $\mathrm{SL}(2,\mathbb{Z})$ action, where

\begin{equation}\label{xiainst}
\widetilde{\xi}_a^{\text{inst}}=\frac{\tau_2}{8\pi^2}\sum_{\hat{\gamma}\in \Lambda^{+}\cup\{0\}}n_{\hat{\gamma}}q_a\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^2}\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m,n}}\,.
\end{equation}

\begin{equation}
\begin{pmatrix}
\end{pmatrix}=\begin{pmatrix}
a &  b\\
c & d\\
\end{pmatrix}^\intercal\begin{pmatrix}
\end{pmatrix},
\end{equation}

where the $\intercal$ subscript denotes the transpose matrix, we find the following transformation rules under $A\in \mathrm{SL}(2,\mathbb{Z})$:

\begin{equation}\label{transrules1}
\tau_2 \to \frac{\tau_2}{|c\tau+d|^2}, \quad |m\tau + n|\to \frac{|m'\tau+ n'|}{|c\tau +d|}, \quad S_{\hat{\gamma},m,n}\to S_{\hat{\gamma},m',n'}\;.
\end{equation}

We now discuss the transformation properties of the factor

\begin{equation}
\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}\,.
\end{equation}

We first consider the case $m\neq 0$. Because $t_{\pm}^{m,n}=t_{\pm}^{km,kn}$ for any $k>0$, we can assume that $(m,n)$ are coprime, so that there exist $p,q\in \mathbb{Z}$ so that

\begin{equation}
\begin{pmatrix}
p & q\\
m & n\\
\end{pmatrix}\in \mathrm{SL}(2,\mathbb{Z})\,.
\end{equation}

Since (<ref>) defines an action on $t$, we find that (making the dependence of $t_{+}^{m,n}$ on $\tau$ explicit)

\begin{equation}
\frac{1+t_{+}^{m', n'}(\tau)t}{t_{+}^{m', n'}(\tau)-t}=\left(\begin{pmatrix}
p & q\\
m & n \\
\end{pmatrix}\cdot \begin{pmatrix}
a & b\\
c & d\\
\end{pmatrix}\right)\cdot t=\begin{pmatrix}
p & q\\
m & n \\
\end{pmatrix}\cdot \left(\begin{pmatrix}
a & b\\
c & d\\
\end{pmatrix}\cdot t\right)=\frac{1+t_{+}^{m,n}\left(\frac{a\tau+b}{c\tau+d}\right)\left(\begin{pmatrix}
a & b\\
c & d\\
\end{pmatrix}\cdot t\right)}{t_{+}^{m,n}\left(\frac{a\tau+b}{c\tau+d}\right)-\begin{pmatrix}
a & b\\
c & d\\
\end{pmatrix}\cdot t}
\end{equation}

so we have the transformation rule

\begin{equation}\label{transrules2}
\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}\to \frac{1+t_{+}^{m', n'}t}{t-t_{+}^{m', n'}}\,.
\end{equation}

The same transformation rule (<ref>) follows when $m=0$ by using the property that $t_{\pm}^{c,d}=t_{\pm}^{kc,kd}$ when $k>0$ and $t_{\pm}^{c,d}=t_{\mp}^{kc,kd}$ when $k<0$, together with (<ref>). Hence, from (<ref>) and (<ref>) it follows that (<ref>) is invariant under the $\mathrm{SL}(2,\mathbb{Z})$-action.

We now verify the transformation rule of $\widetilde{\xi}_0^{\text{inst}}$ in (<ref>). For this we use that under the $\mathrm{SL}(2,\mathbb{Z})$-action:

\begin{equation}\label{transrules3}
m\xi^0+n \to \frac{m'\xi^0+ n'}{c\xi^0+d}, \quad \frac{m\tau_1+ n}{|m\tau + n|^2}\to \frac{c(m'|\tau|^2 +n'\tau_1)+d(m'\tau_1+n')}{|m'\tau+n'|^2}\,.
\end{equation}

After a rather lengthy but straightforward computation, one can also check that

\begin{equation}\label{transrules5}
\frac{1-t_{+}^{m,n}t}{t-t_{+}^{m, n}}t^a\to (c\tau_1+d)\frac{1-t_{+}^{m', n'}t}{t-t_{+}^{m', n'}}t^a-c\frac{t+t_{+}^{m', n'}}{t-t_{+}^{m', n'}}\tau_2t^a\;.
\end{equation}

We therefore find using (<ref>), (<ref>), (<ref>) and (<ref>), that the terms of $\widetilde{\xi}_0^{\text{inst}}$ transform as follows

\begin{equation}
\begin{split}
&\frac{\I \tau_2}{16\pi^3}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}^{(0)}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(\frac{1}{m\xi^0+n}+\frac{m\tau_1+ n}{|m\tau+ n|^2}\right)\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau + n|^2}\\
&\to d\left(\frac{\I \tau_2}{16\pi^3}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}^{(0)}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(\frac{1}{m\xi^0+n}+\frac{m\tau_1+ n}{|m\tau+ n|^2}\right)\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau + n|^2}\right)\\
&\quad\quad\quad\quad+c\left(\frac{\I\tau_2}{16\pi^3}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}^{(0)}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(\frac{\xi^0}{m\xi^0+ n}+\frac{m|\tau|^2+ n\tau_1}{|m\tau+ n|^2}\right)\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau + n|^2}\right)\;,\\
\end{split}
\end{equation}

\begin{equation}
\begin{split}
&-\frac{\tau_2}{8\pi^2}\sum_{\hat{\gamma}\in \Lambda_{+}\cup \{0\}}n_{\hat{\gamma}} \sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(q_ab^a \frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}+\I q_at^a\frac{1-t_{+}^{m, n}t}{t-t_{+}^{m, n}}\right)\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^2}\\
&\to d\left(-\frac{\tau_2}{8\pi^2}\sum_{\hat{\gamma}\in \Lambda_{+}\cup \{0\}}n_{\hat{\gamma}} \sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(q_ab^a \frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}+\I q_at^a\frac{1-t_{+}^{m, n}t}{t-t_{+}^{m, n}}\right)\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^2}\right)\\
&\quad\quad+c\left(-\frac{\tau_2}{8\pi^2}\sum_{\hat{\gamma}\in \Lambda_{+}\cup \{0\}}n_{\hat{\gamma}}^{(0)} \sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(q_ac^a \frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}+\I q_at^a\left(\tau_1\frac{1-t_{+}^{m, n}t}{t-t_{+}^{m, n}}-\tau_2\frac{t+t_{+}^{m, n}}{t-t_{+}^{m, n}}\right)\right)\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^2}\right)
\end{split}
\end{equation}

so overall

\begin{equation}
\widetilde{\xi}_0^{\text{inst}}\to d\widetilde{\xi}_0^{\text{inst}}-c\widetilde{\alpha}^{\text{inst}}\,.
\end{equation}

Finally, we check the transformation rule for $\widetilde{\alpha}^{\text{inst}}$ in (<ref>). Given that we know the transformation rules of the rest of the variables, it is easy to check that this is equivalent to showing that $\alpha^{\text{inst}}=\widetilde{\alpha}^{\text{inst}}+\xi^i\widetilde{\xi}_i^{\text{inst}}$ transforms by

\begin{equation}
\alpha^{\text{inst}}\to \frac{\alpha^{\text{inst}}}{c\xi^0+d}\,.
\end{equation}

To check the later, one uses the fact that under the action of $A\in \mathrm{SL}(2,\mathbb{Z})$

\begin{equation}\label{transrules4}
(m\tau_1+n)(t^{-1}-t)-2m \tau_2 \to \frac{(m'\tau_1+ n')(t^{-1}-t)-2m'\tau_2}{c\xi^0+d}\,.
\end{equation}

The result then follows immediately from the formula for $\alpha^{\text{inst}}$ obtained via (<ref>) and (<ref>), and the transformations (<ref>), (<ref>).

Let $(\widetilde{N},g_{\overline{N}})$ be an instanton corrected q-map space with $1$-loop parameter $c_{\ell}=\frac{\chi}{192\pi}$.
If, after possibly restricting $\widetilde{N}$,  we have that $\widetilde{N}$ is invariant under the action of $A\in \mathrm{SL}(2,\mathbb{Z})$ by the S-duality action (<ref>), then $A$ also acts by isometries on $(\widetilde{N},g_{\overline{N}})$. In particular, if $\widetilde{N}$ is invariant under the full S-duality action, then $\mathrm{SL}(2,\mathbb{Z})$ acts by isometries on $(\widetilde{N},g_{\overline{N}})$.

Since $A$ leaves $\widetilde{N}$ invariant, we have $S_A:\mathcal{Z}\to \mathcal{Z}$ the diffeomorphism obtained by lifting the S-duality action of $A$ to the twistor space $\mathcal{Z}\cong \widetilde{N}\times \mathbb{C}P^1$ by (<ref>). We want to show that $S_A$ is a twistor space automorphism (i.e. it is holomorphic, preserves the contact distribution, and commutes with the real structure of the twistor space), so that we can conclude that the action of $A$ on $(\widetilde{N},g_{\overline{N}})$ is isometric.

We first want to show that $S_A$ is holomorphic. Notice that if $S^1\subset \mathbb{C}P^1$ denotes the compactification of the real line $\mathbb{R}\subset \mathbb{C}$ (with respect to the $t$-coordinate), the Darboux coordinates $(\xi^i,\widetilde{\xi}_i,\widetilde{\alpha})$ are defined on the open dense subset of $\mathcal{Z}$ given by

\begin{equation}
\mathcal{Z}_D:=\mathcal{Z}-\widetilde{N}\times S^1,
\end{equation}
since for each $p\in \widetilde{N}$, the singularities of the Darboux coordinates are at $t=0,\infty$ and $t\in \{t_+^{m,n}\}_{m\neq 0,n\in \mathbb{Z}}\subset \mathbb{R}\subset S^1$. Using the coordinates $(\xi^i,\widetilde{\xi}_i,\widetilde{\alpha})$ on $\mathcal{Z}$, and using that Darboux coordinates for a holomorphic contact structure must be holomorphic coordinates (see for example the proof of the second statement of <cit.>), the transformation rule (<ref>) shows that the map

\begin{equation}
S_A:\mathcal{Z}_D\cap S^{-1}_A(\mathcal{Z}_D)\to \mathcal{Z}_D
\end{equation}

is holomorphic.
On the other hand, if $\mathcal{I}$ denotes the holomorphic structure of $\mathcal{Z}$, we have that the diffeomorphism $S_A$ is holomorphic if and only if

\begin{equation}\label{holcond}
\mathcal{I}\circ \D S_A=\D S_A \circ \mathcal{I}\,.
\end{equation}

Since $\mathcal{Z}_D\cap S^{-1}_A(\mathcal{Z}_D)$ is the intersection of two open dense subsets of $\mathcal{Z}$, we have that $\mathcal{Z}_D\cap S^{-1}_A(\mathcal{Z}_D)$ must be dense in $\mathcal{Z}$. We then have that (<ref>) holds on a dense set of $\mathcal{Z}$. Since $S_A$ and $\mathcal{I}$ are globally defined on $\mathcal{Z}$, we then conclude by continuity that (<ref>) must hold on all of $\mathcal{Z}$, and $S_A$ must be holomorphic.

On the other hand, the fact that the coordinates $(\xi^i,\widetilde{\xi}_i,\widetilde{\alpha}=\alpha-\xi^i\widetilde{\xi}_i)$ transform via (<ref>) implies that

\begin{equation}
S_A^*(\D \widetilde{\alpha} + \xi^i\D \widetilde{\xi}_i)=\frac{\D \widetilde{\alpha} + \xi^i\D \widetilde{\xi}_i}{c\xi^0+d}\,.
\end{equation}

Hence $S_A$ preserves the contact distribution $\text{Ker}(\lambda)$ on a dense subset of $\mathcal{Z}$.
We then conclude as before by continuity and the fact that the contact distribution is globablly defined, that the contact distribution must be globally preserved by $S_A$.

Finally, to check that the action of $A$ preserves the real structure, it is enough to check that (<ref>) commutes with the antipodal map $t \to -1/\overline{t}$. Indeed, we have for $c\neq 0$ that

\begin{equation}
\begin{pmatrix}
a & b\\
c & d\\
\end{pmatrix}\cdot \Big(-\frac{1}{\overline{t}}\Big)=\frac{\overline{t}-t_{+}^{c,d}}{t_+^{c,d}\overline{t}+1}=-\overline{\Big[\begin{pmatrix}
a & b\\
c & d\\
\end{pmatrix}\cdot t \Big]}^{-1}\,,
\end{equation}

where we have used that $t_{+}^{c,d}\in \mathbb{R}$. The case when $c=0$ follow by a trivial computation.

Hence, we conclude that the action of $A$ is via twistor space automorphisms, and hence $\mathrm{SL}(2,\mathbb{Z})$ must act via isometries on $(\widetilde{N},g_{\overline{N}})$.

For the final statement, if $\widetilde{N}$ is invariant under the $\mathrm{SL}(2,\mathbb{Z})$ S-duality action, then we can lift the $\mathrm{SL}(2,\mathbb{Z})$ action to the twistor space via (<ref>). By the proof of the first statement, this lift acts by twistor space automorphisms, so $\mathrm{SL}(2,\mathbb{Z})$ must act by isometries on $(\widetilde{N},g_{\overline{N}})$.

Recall that $\mathrm{SL}(2,\mathbb{Z})$ is generated by $T$ and $S$ given in (<ref>). The transformations generated by $T$ correspond to part of the usual Heisenberg isometries (see Section <ref> below). On the other hand, the transformation given by $S$ in the “non-trivial" transformation that exchanges weak and strong coupling in the type IIB string theory setting. The following theorem guarantees that an instanton corrected q-map space always carries (after possibly restricting $\widetilde{N}$), an action by isometries by $S\in \mathrm{SL}(2,\mathbb{Z})$.

Let $S\in \mathrm{SL}(2,\mathbb{Z})$ be given as in (<ref>), and consider an instanton corrected q-map space $(\widetilde{N},g_{\overline{N}})$  with $c_{\ell}=\frac{\chi}{192\pi}$. Then we can find a non-empty $S$-invariant open subset $\widetilde{N}_S\subset \widetilde{N}$ such that the restricted instanton corrected q-map space $(\widetilde{N}_S,g_{\overline{N}})$ is positive definite and carries a $\mathbb{Z}_4$-action by isometries generated by $S$. The open subset $\widetilde{N}_S$ is given by

\begin{equation}\label{Sinvariantsub}
\widetilde{N}_S=\{p\in \widetilde{N}\; | \; \epsilon<\tau_2, \;\; \epsilon < \frac{\tau_2}{|\tau_1|^2+|\tau_2|^2}, \;\; t^a>K, \;\; |\tau|t^a> K \}
\end{equation}

for some $0<\epsilon<1$ and $K>0$.

We would first like to show that there is $K>0$ and $0<\epsilon<1$ such that $g_{\overline{N}}$ is defined and positive definite on

\begin{equation}
\widetilde{N}_{K,\epsilon}:=\{(\tau_2,b^a+\mathrm{i}t^a,\zeta^i,\widetilde{\zeta}_i,\sigma)\in \widetilde{N} \; | \; \tau_2>\epsilon, \;\; t^a>K, \;\; a=1,...,n\}\,.
\end{equation}

In order to show this we first study the CASK geometry of signature $(2,2n)$ defined by $(M,-\mathfrak{F})$. In terms of the natural holomorphic coordinates $Z^i$, $i=0,...,n$, the CASK geometry has a Kähler potential $(Z^i,\overline{Z}^i)$ given by

\begin{equation}
k(Z^i,\overline{Z}^i)=-\text{Im}(\tau_{ij})Z^i\overline{Z}^j=-|Z^0|^2\text{Im}(\tau_{ij})z^i\overline{z}^j\,, \quad z^i=\frac{Z^i}{Z^0}\,.
\end{equation}

Since $Z^0\neq 0$ on $M$, we can use instead the holomorphic coordinates $(Z^0,z^a)$. In terms of $(Z^0,z^a)$ we find using the formula (<ref>) for $\mathfrak{F}$, that

\begin{equation}\label{KpotCASK}
\begin{split}
&=|Z^0|^2\left(4h(t)-\frac{\chi \zeta(3)}{(2\pi)^3}+\frac{2}{(2\pi)^3}\sum_{q_a\gamma^a \in \Lambda^+}n_{\gamma}\text{Re}(\mathrm{Li}_3(e^{2\pi iq_az^a}))+\frac{2}{(2\pi )^2}\sum_{q_a\gamma^a \in \Lambda^+}n_{\gamma}\text{Re}(\mathrm{Li}_2(e^{2\pi iq_az^a}))q_at^a\right)\,,\\
\end{split}
\end{equation}

where we recall that $h(t)$ is the cubic polynomical defining the PSR manifold (see Section <ref>). From the previous formula, it immediately follows that the coefficients of the CASK metric $g_M$ in the coordinates $(Z^0,z^a)$ only depend periodically on $b^a=\text{Re}(z^a)$. Furthermore, as $t^a=\text{Im}(z^a)\to \infty$ the classical terms due to $\mathfrak{F}^{\text{cl}}$ dominate over the terms due to $\mathfrak{F}^{\text{w.s}}$, which are either independent of $t^a$, or exponentially decreasing as $t^a\to \infty$. Since the classical terms must satisfy the CASK conditions, we find that there is $K>0$, such that

\begin{equation}
M_K:=\{ (Z^0,b^a+\I t^a)\in M^q  \; | \; t^a>K, \;\; a=1,...,n\}\subset M\,,
\end{equation}

where $M$ was defined in Section <ref> as the maximal open set of $M^{\text{cl}}$ where the CASK geometry defined by $(M,\mathfrak{F})$ has signature $(2n,2)$ and $\text{Im}(\tau_{ij})Z^i\overline{Z}^j<0$.

Now let us look at the tensor $T$ defined in (<ref>), determining the compatibility condition between the CASK structure and the BPS structure. The instanton constribution to $T$ due to the BPS indices is by an expression of the form

\begin{equation}
\sum_{\gamma}\Omega(\gamma)\sum_{n>0}e^{2\pi \I n\zeta_{\gamma}}K_0(2\pi n \tau_2|q_0+q_a(b^a+\I t^a)|)|\D Z_{\gamma}|^2, \quad \gamma=q_0\gamma^0+q_a\gamma^a\,.
\end{equation}

Due to the exponential decay of the Bessel functions $K_0(x)$ as $x\to \infty$ and the convergence property of the BPS structure, by bounding below $\tau_2$ and increasing $K$ if necessary, we can make the CASK metric term of (<ref>) dominate over the instanton corrections uniformly in the rest of the parameters, and in particular make $T$ horizontally non-degenerate on this region. More precisely, if $0<\epsilon<1$, there is $K>0$ sufficiently big such that the (pseudo-)HK metric $g_{N}$ is defined on

\begin{equation}
N_{K,\epsilon}:=\{ (Z^0,z^a,\zeta^i,\widetilde{\zeta}_i)\in M\times (\mathbb{R}/\mathbb{Z})^{2n+2} \; | \; \epsilon<\tau_2=|Z^0|,\;\; t^a>K, \; a=1,...,n\}\,.
\end{equation}

By the same arguments as those given in the previous paragraphs, we find that as $t^a\to \infty$ for $a=1,...,n$, the functions $f$, $f_3$ and $g_{N}(V,V)$ defined on Section <ref> are asymptotically approximated  (uniformly in the other parameters) by $f^{\text{cl}}$, $f_3^{\text{cl}}$ and $g_{N}^{\text{cl}}(V,V)$, where the superscript $^{\text{cl}}$ refers to the corresponding functions obtained by setting $n_{\hat{\gamma}}=0$ for all $\hat{\gamma}\in \Lambda^{+}\cup\{0\}$. Since we have $f^{\text{cl}}>0$, $f_3^{\text{cl}}<0$ and $g_{N}^{\text{cl}}(V,V)\neq 0$, it follows that we can pick $K$ such that on $N_{K,\epsilon}$ we obtain that

\begin{equation}
f>0, \quad f_3 <0, \quad  g_N(V,V) \neq 0\,.
\end{equation}

This ensures by the end of Theorem <ref> that $g_{\overline{N}}$ is defined and positive definite on

\begin{equation}
\overline{N}_{K,\epsilon}:=\{(\tau_2,b^a+\mathrm{i}t^a,\zeta^i,\widetilde{\zeta}_i,\sigma)\in \overline{N} \; | \; \tau_2>\epsilon, \;\; t^a>K, \; a=1,...,n\}\,.
\end{equation}

As before, we lift the metric $g_{\overline{N}}$ to the subset $\widetilde{N}_{K,\epsilon}\to N_{K,\epsilon}$ where the periodic coordinates are made non-periodic.

Now notice that since $S^4=\text{Id}$, the open subset

\begin{equation}\label{Sinvariantint}
\widetilde{N}_S:=\widetilde{N}_{K,\epsilon}\cap S\cdot \widetilde{N}_{K,\epsilon} \cap S^2\cdot \widetilde{N}_{K,\epsilon}\cap S^3\cdot \widetilde{N}_{K,\epsilon}
\end{equation}

is $S$-invariant. To see that it is also non-empty, notice that the points (in Type IIB coordinates) of the form $(\tau_1+\I\tau_2,b^a+\I t^a, c^a,c_a,c_0,\psi)=(0+\I, 0 +\I t^a, 0 ,0,0,0)$ are $S$-fixed. In particular, for $t^a>K$ we have $(0+\I, 0 +\I t^a, 0 ,0,0,0)\in \widetilde{N}_{\epsilon,K}$ and, since these points are $S$-fixed, they must lie on $\widetilde{N}_S$. Since $S$ acts on $\widetilde{N}_S$, we can apply Theorem <ref> in the case of $A=S\in \mathrm{SL}(2,\mathbb{Z})$ to concluce that $(\widetilde{N}_S,g_{\overline{N}})$ carries a $\mathbb{Z}_4$-action by isometries generated by $S$.

Finally, the fact that $\widetilde{N}_S$ is given by (<ref>) follows immediately from the defining relations of $\widetilde{N}_{K,\epsilon}$ together with (<ref>) and the S-duality transformations (<ref>) which imply that $S^2\cdot  \widetilde{N}_{K,\epsilon}=\widetilde{N}_{K,\epsilon}$.

§ UNIVERSAL ISOMETRIES OF INSTANTON CORRECTED Q-MAP SPACES AND S-DUALITY

We start by recalling a certain universal group of isometries in the case of a tree-level q-map space (recall Definition <ref>).
In <cit.> it was shown that a tree level q-map space of real dimension $4n+4$ with $n>0$ has a universal (i.e. independent of the PSR manifold) $3n+6$ dimensional connected Lie group of isometries $G$, whose Lie algebra $\mathfrak{g}$ has the form

\begin{equation}\label{uniisotree}
\mathfrak{g}=\mathbb{R}\ltimes (\mathfrak{sl}(2,\mathbb{R})\ltimes (\mathbb{R}^n\ltimes \mathfrak{h}_{2n+2}))\,.
\end{equation}

The first factor corresponds to an action by dilations of the group $\mathbb{R}_{>0}$; the second to the S-duality $\mathrm{SL}(2,\mathbb{R})$-action; the third to an action by $\mathbb{R}^n$ which, among other things, shifts the real part of the PSK coordinates $z^a$; and the last factor to the action of a certain codimension $1$ subgroup $H_{2n+2}$ of the Heisberg group $\text{Heis}_{2n+3}(\mathbb{R})$ (to be defined below). In order to state this more precisely, we consider the coordinates $(\rho^{\text{cl}},z^a,\zeta^i,\widetilde{\zeta}_i^{\text{cl}},\sigma^{\text{cl}})$ on $\overline{\mathcal{N}}_{\text{IIA}}^{\text{cl}}:=\mathbb{R}_{>0}\times \overline{M}^{\text{cl}}\times \mathbb{R}^{2n+2}\times \mathbb{R}$, related to the type IIB coordinates $(\tau,b^a+\mathrm{i}t^a,c^a,c_a,c_0,\psi)$ on $\overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}$ (where the latter was given in Definition <ref>) via the classical Mirror map (i.e. by (<ref>) with $c_{\ell}=n_{\hat{\gamma}}=\chi=0$). The groups then act as follows:

* The multiplicative $\mathbb{R}_{>0}$ group acts by a dilation action on the $(\rho^{\text{cl}},\zeta^i,\widetilde{\zeta}_i^{\text{cl}},\sigma^{\text{cl}})$ variables via

\begin{equation}\label{scalingaction}
r\cdot (\rho^{\text{cl}},\zeta^i,\widetilde{\zeta}_i^{\text{cl}},\sigma^{\text{cl}})=(r\rho^{\text{cl}}, \sqrt{r}\zeta^i,\sqrt{r}\widetilde{\zeta}_i^{\text{cl}},r\sigma^{\text{cl}})\,.
\end{equation}

* The $\mathrm{SL}(2,\mathbb{R})$ factor corresponds to the S-duality action given in (<ref>). We have $\mathrm{SL}(2,\mathbb{R})$ instead of $\mathrm{SL}(2,\mathbb{Z})$ due to the absence of quantum corrections.

* The vector $v=(v^a)\in \mathbb{R}^n$ acts via

\begin{equation}\label{Rnaction}
v\cdot \begin{pmatrix}
\rho^{\text{cl}} \\
\zeta^0 \\
\zeta^a\\
\widetilde{\zeta}_0^{\text{cl}}\\
\widetilde{\zeta}_a^{\text{cl}}\\
\sigma^{\text{cl}} \\
\end{pmatrix}=\begin{pmatrix}
\rho^{\text{cl}} \\
\zeta^0 \\
\zeta^a +\zeta^0v^a\\
\widetilde{\zeta}_0^{\text{cl}}+\frac{1}{6}k_{abc}v^av^bv^c\zeta^0 + \frac{1}{2}k_{abc}v^av^b\zeta^c -\widetilde{\zeta}_a^{\text{cl}}v^a\\
\widetilde{\zeta}_a ^{\text{cl}}-\frac{1}{2}\zeta^0k_{abc}v^bv^c - k_{abc}v^b\zeta^c\\
\sigma^{\text{cl}} \\
\end{pmatrix}\,,
\end{equation}

where we recall that $k_{abc}$ are the coefficients of the cubic polynomial (<ref>) defining the PSR manifold.

* If $\text{Heis}_{2n+3}(\mathbb{R})\cong \mathbb{R}^{2n+3}$ denotes the Heisenberg group, then $(\eta^i,\widetilde{\eta}_i,\kappa)\in \text{Heis}_{2n+3}(\mathbb{R})$, $i=0,1,...,n$; acts on $(\zeta^i,\widetilde{\zeta}_i^{\text{cl}},\sigma^{\text{cl}})$ via

\begin{equation}\label{heisact}
(\eta^i,\widetilde{\eta}_i,\kappa)\cdot (\zeta^i,\widetilde{\zeta}_i^{\text{cl}},\sigma^{\text{cl}})=(\zeta^i+\eta^i, \widetilde{\zeta}_i^{\text{cl}}+\widetilde{\eta}_i,  \sigma^{\text{cl}} +\kappa +\widetilde{\zeta}_i^{\text{cl}}\eta^i-\zeta^i\widetilde{\eta}_i)\,.
\end{equation}

On the other hand, $H_{2n+2}\subset \text{Heis}_{2n+3}(\mathbb{R})$ is the codimension $1$ subgroup given by

\begin{equation}\label{codim1heis}
H_{2n+2}:=\{(\eta^i,\widetilde{\eta}_i,\kappa) \in \text{Heis}_{2n+3}(\mathbb{R}) \; | \; \eta^0=0\}\,.
\end{equation}

The transformations shifting $\zeta^0$ missing from $H_{2n+2}$ are already included in the $\mathrm{SL}(2,\mathbb{R})$ transformations.

On the other hand, the semi-direct product of Lie algebras $\mathbb{R}^n\ltimes \mathfrak{h}_{2n+2}$ in (<ref>) corresponds at the group level to the semi-direct product $\mathbb{R}^n\ltimes_{\varphi} H_{2n+2}\subset \mathbb{R}^n\ltimes_{\varphi} \text{Heis}_{2n+3}(\mathbb{R})$, where the automorphism $\varphi:\mathbb{R}^n\to \text{Aut}(\text{Heis}_{2n+3}(\mathbb{R}))$ is given by a similar formula to (<ref>), namely

\begin{equation}\label{autheis}
\varphi(v)\cdot \begin{pmatrix}

\eta^0 \\
\eta^a\\
\widetilde{\eta}_0\\
\widetilde{\eta}_a\\
\kappa \\
\end{pmatrix}=\begin{pmatrix}
\eta^0 \\
\eta^a +\eta^0v^a\\
\widetilde{\eta}_0+\frac{1}{6}k_{abc}v^av^bv^c\eta^0 + \frac{1}{2}k_{abc}v^av^b\eta^c -\widetilde{\eta}_av^a\\
\widetilde{\eta}_a -\frac{1}{2}\eta^0k_{abc}v^bv^c - k_{abc}v^b\eta^c\\
\kappa \\
\end{pmatrix}, \quad v\in \mathbb{R}^n, \quad (\eta^i,\widetilde{\eta}_i,\kappa)\in \text{Heis}_{2n+3}(\mathbb{R})\,.
\end{equation}

The other semi-direct products of Lie algebras in (<ref>) can be described via the Lie algebra structure given in <cit.>.

We now consider $(M,\mathfrak{F})$ and a mutually local variation of BPS structures $(M,\Gamma,Z,\Omega)$ as in Section <ref>, so that we obtain an instanton corrected q-map space $(\widetilde{N},g_{\overline{N}})$. We want to study how to instanton corrections affect the $3n+6$-dimensional isometry group $G$ from the tree-level q-map space case. In the following, unless otherwise specified, we assume that $\widetilde{N}$ is the (lift of the) maximal domain of definition of $g_{\overline{N}}$ obtained via HK/QK correspondence from the HK metric $(N,g_{N})$ from Section <ref>. We also define the following subgroups of $\text{Heis}_{2n+3}(\mathbb{R})$:

\begin{equation}\label{heisinstgroup}
\begin{split}
\text{Heis}_{2n+3,D}&:=\{ (\eta^i,\widetilde{\eta}_i,\kappa)\in \text{Heis}_{2n+3}(\mathbb{R}) \; |\quad \; \text{$\eta^i \in \mathbb{Z}$ for $i=0,...,n$}\}\\
H_{2n+2,D}&:=\{ (\eta^a,\widetilde{\eta}_i,\kappa)\in H_{2n+2} \; |\quad \; \text{$\eta^a \in \mathbb{Z}$ for $a=1,...,n$}\}\,.
\end{split}
\end{equation}

The letter $D$ in the above notation is meant to emphasize that the directions $\eta^i$ are broken to a discrete subgroup due to the inclusion of (part of) the “D-instanton corrections" due to terms involving the BPS indices $\Omega(\gamma)$.

We begin by studying how $\text{Heis}_{2n+3,D}\subset \text{Heis}_{2n+3}(\mathbb{R})$ and $\mathbb{Z}^n\subset \mathbb{R}^n$ act on the corrected coordinates $(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)$, related to the type IIB coordinates via the quantum corrected mirror map (<ref>). We use the notation $\rho^{\text{w.s.}}:=f^{\text{w.s.}}/16\pi$, so that $\rho=\rho^{\text{cl}}+\rho^{\text{w.s.}}-c_{\ell}$ (recall (<ref>) and (<ref>)).

$\text{Heis}_{2n+3,D}$ acts on the functions $(\rho^{\text{w.s.}},\widetilde{\zeta}_i^{\text{inst}},\sigma^{\text{inst}})$ by

\begin{equation}\label{heisinst}
(\eta^i,\widetilde{\eta}_i,\kappa)\cdot (\rho^{\text{w.s.}},\widetilde{\zeta}_i^{\text{inst}},\sigma^{\text{inst}})=(\rho^{\text{w.s.}},\widetilde{\zeta}_i^{\text{inst}},\sigma^{\text{inst}}+\widetilde{\zeta}_i^{\text{inst}}\eta^i)\,.
\end{equation}

On the other hand, $\mathbb{Z}^n\subset \mathbb{R}^n$ acts on the functions $(\rho^{\text{w.s.}},\widetilde{\zeta}_i^{\text{inst}},\sigma^{\text{inst}})$ by

\begin{equation}\label{zninst}
(v^a)\cdot (\rho^{\text{w.s.}},\widetilde{\zeta}_0^{\text{inst}},\widetilde{\zeta}_a^{\text{inst}},\sigma^{\text{inst}})=(\rho^{\text{w.s.}},\widetilde{\zeta}_0^{\text{inst}}-v_a\widetilde{\zeta}_a^{\text{inst}},\widetilde{\zeta}_a^{\text{inst}},\sigma^{\text{inst}})
\end{equation}

The first statement (<ref>) follows easily from (<ref>), (<ref>) and (<ref>). On the other hand note that (<ref>) and the fact that $\rho^{\text{cl}}=\frac{\tau_2^2}{2}h(t)$ imply that $\tau_2$ is invariant under the action of $\mathbb{Z}^n$. Equation (<ref>) then follows again easily from (<ref>) and (<ref>). The restrictions to $\text{Heis}_{2n+3,D}\subset \text{Heis}_{2n+3}(\mathbb{R})$ and $\mathbb{Z}^n\subset \mathbb{R}^n$ are required, since in the computation we use the periodicity of the complex exponential function.

The action of $\text{Heis}_{2n+3,D}\subset \text{Heis}_{2n+3}(\mathbb{R})$ and $\mathbb{Z}^n\subset \mathbb{R}^n$ on $(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)\in \overline{\mathcal{N}}_{\text{IIA}}=\mathbb{R}_{>-c_{\ell}}\times \overline{M}\times \mathbb{R}^{2n+2}\times \mathbb{R}$ has the same transformation rules, (<ref>) and (<ref>), as their action on $(\rho^{\text{cl}},z^a,\zeta^i,\widetilde{\zeta}_i^{\text{cl}},\sigma^{\text{cl}})\in \overline{\mathcal{N}}_{\text{IIA}}^{\text{cl}}$. Namely, for $v\in \mathbb{Z}^n$ and $(\eta^i,\widetilde{\eta}_i,\kappa)\in \text{Heis}_{2n+3,D}$,

\begin{equation}\label{ZnHeisact}
v\cdot \begin{pmatrix}
\rho \\
\zeta^0 \\
\zeta^a\\
\widetilde{\zeta}_0\\
\widetilde{\zeta}_a\\
\sigma \\
\end{pmatrix}=\begin{pmatrix}
\rho \\
\zeta^0 \\
\zeta^a +\zeta^0v^a\\
\widetilde{\zeta}_0+\frac{1}{6}k_{abc}v^av^bv^c\zeta^0 + \frac{1}{2}k_{abc}v^av^b\zeta^c -\widetilde{\zeta}_a^{\text{cl}}v^a\\
\widetilde{\zeta}_a -\frac{1}{2}\zeta^0k_{abc}v^bv^c - k_{abc}v^b\zeta^c\\
\sigma \\
\end{pmatrix}\,, \quad (\eta^i,\widetilde{\eta}_i,\kappa)\cdot \begin{pmatrix}z^a\\
\rho\\
\zeta^i\\
\widetilde{\zeta}_i\\
\sigma\end{pmatrix}=\begin{pmatrix}z^a\\
\rho\\
\zeta^i+\eta^i\\ \widetilde{\zeta}_i+\widetilde{\eta}_i\\
\sigma +\kappa +\widetilde{\zeta}_i\eta^i-\zeta^i\widetilde{\eta}_i\\
\end{pmatrix}\,.
\end{equation}

Furthermore, the action of $\text{Heis}_{2n+3,D}$ and $\mathbb{Z}^n$ on $(z^a,\rho,\zeta^i, \widetilde{\zeta}_i,\sigma)$ expressed in terms of type IIB coordinates via the quantum corrected mirror map (<ref>) coincides with the action (<ref>) and (<ref>) expressed in type IIB variables via the classical mirror map.

The first statement follow from Lemma <ref> together with the actions (<ref>) and (<ref>). The last statement follows immediately from (<ref>), together with (<ref>), (<ref>) and (<ref>).

* We remark that since the actions (<ref>) leave the $t^a=\text{Im}(z^a)$ coordinates invariant, via the quantum corrected mirror map we get an action of $\mathbb{Z}^n$ and $\text{Heis}_{2n+3,D}$ not only on  $\overline{\mathcal{N}}_{\text{IIB}}=\mathcal{M}^{-1}(\overline{\mathcal{N}}_{\text{IIA}})$, but also the bigger manifold $\overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}\supset \overline{\mathcal{N}}_{\text{IIB}}$ where $\mathrm{SL}(2,\mathbb{Z})$ always acts via (<ref>).

* It is not hard to the check that $(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)$ do not have any nice transformation property under the $\mathbb{R}_{>0}$ scaling action (<ref>).

The group $\text{Heis}_{2n+3,D}$ acts by isometries on any instanton corrected c-map space $(\widetilde{N},g_{\overline{N}})$ (recall Definition <ref>). Furthermore, if $n_{\hat{\gamma}}=0$ for all $\hat{\gamma}\in \Lambda^{+}$ (but with $\chi$ possibly non-zero), then $(\widetilde{N},g_{\overline{N}})$ carries an action by isometries of $\{(\eta^i,\widetilde{\eta}_i,\kappa)\in \text{Heis}_{2n+3}(\mathbb{R})\; | \; \eta^0\in \mathbb{Z}\}$, and in particular by $H_{2n+2}$.

By Corollary <ref>, the group $\text{Heis}_{2n+3,D}$ acts on the $(\zeta^i,\widetilde{\zeta}_i,\sigma)$ part of the type IIA coordinates $(\rho, z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)$ via (<ref>). Because the instanton corrections are periodic in $\zeta^i$ and not depend on $\widetilde{\zeta}_i$ and $\sigma$, it follows immediately that the tensor $T$ in (<ref>) as well as the sets $N'$ from (<ref>) and $\overline{N}$ from (<ref>) are invariant under the action of $\text{Heis}_{2n+3,D}$. In particular, it follows that $\widetilde{N}$ must carry an action of $\text{Heis}_{2n+3,D}$. On the other hand, by the explicit formula (<ref>) for $g_{\overline{N}}$ and using again the fact that all instanton correction terms are periodic in $\zeta^i$ and independent from $\widetilde{\zeta}_i$ and $\sigma$, it can be easily checked that  the action (<ref>) of $\text{Heis}_{2n+3,D}$ acts by isometries on $(\widetilde{N},g_{\overline{N}})$.

The last statement follows immediately from the previous argument, together with the fact that if $n_{\hat{\gamma}}=0$ for all $\hat{\gamma}\in \Lambda^+$, then the instanton correction terms only depend on $\zeta^0$.

We now show the following proposition, dealing with the $\mathbb{R}^n$-factor of (<ref>) in the case of instanton corrected q-map spaces.

Let $(\widetilde{N},g_{\overline{N}})$ be an instanton corrected q-map space. Then $(\widetilde{N},g_{\overline{N}})$ carries an action by isometries by $\mathbb{Z}^n\subset \mathbb{R}^n$ given by (<ref>). Furthermore, if $n_{\hat{\gamma}}=0$ for all $\hat{\gamma}\in \Lambda^{+}$ (but with $\chi$ possibly non-zero), then $(\widetilde{N},g_{\overline{N}})$ carries an action by isometries by the full $\mathbb{R}^n$.

Notice that with respect to the coordinates $(Z^0,z^a)=(Z^0,Z^a/Z^0)$ of $M$, we can write the Kähler potential $k$ of the CASK manifold as (<ref>). In particular, we see that $k(Z^0,b^a+\I t^a)$ is invariant under integer shifts of the $b^a$, and hence, $M$ must be invariant under the action of $\mathbb{Z}^n$, and it must be an isometry of the CASK metric.

On the other hand, when considering the associated instanton corrected HK geometry, one must consider the tensor $T$ given in (<ref>). In our particular case, we can rewrite $T$ as follows

\begin{equation}\label{Tqmap}
\begin{split}
T&=-\text{Im}(\tau_{ij})\D Z^i \D\overline{Z}^j +\frac{\chi}{2\pi}\sum_{q_0\in \mathbb{Z}-\{0\}}\sum_{n>0}e^{-2\pi \mathrm{i}nq_0\zeta^0}K_0(4\pi R n|q_0|)|q_0\D Z^0|^2 \\
&\quad \quad -\frac{1}{2\pi}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\sum_{q_0\in \mathbb{Z}}\sum_{n>0}e^{-2\pi \mathrm{i}n(q_a\zeta^a +q_0\zeta^0)}K_0(4\pi R n|q_0+q_az^a|)|Z^0q_a\D z^a+(q_0+q_az^a)\D Z^0|^2\,.
\end{split}
\end{equation}

From the above formula together with (<ref>), it follows that the instanton part of $T$ in (<ref>) is invariant under the action of $\mathbb{Z}^n$. Indeed, each summand in the term proportional to $\chi$ is invariant, while in the last term one finds that for each fixed $\hat{\gamma}=q_a\gamma^a\in \Lambda^{+}$, we have $q_0\to q_0+q_av^a$, which remains invariant since we have a sum over all $q_0\in \mathbb{Z}$.  Since $\mathbb{Z}^n$ also acts by isometries on the CASK metric, it follows that $T$ is invariant under $\mathbb{Z}^n$, and hence the maximal domain of definition $N$ of the HK metric is invariant under $\mathbb{Z}^n$. Similarly, the conditions defining the subsets $N'$ in (<ref>) and $\overline{N}$ in (<ref>) required in the construction of the instanton corrected QK manifold via HK/QK correspondence are seen to be $\mathbb{Z}^n$-invariant, so that $\widetilde{N}$ carries an action of $\mathbb{Z}^n$. To show that the action of $\mathbb{Z}^n$ is by isometries, we show that one can lift the action to the twistor space acting by twistor space automorphisms. We define the lift by declaring that it acts trivially on the $\mathbb{C}P^1$ fibers of $\mathcal{Z}\cong \widetilde{N}\times \mathbb{C}P^1$. By using the explicit formulas for $(\xi^i,\widetilde{\xi}_i^{\text{cl}},\alpha^{\text{cl}})$ in (<ref>) one finds that they have the following transformation rule under the action of $\mathbb{Z}^n$ (or even $\mathbb{R}^n$):

\begin{equation}
\xi^0\\
\xi^a\\
\widetilde{\xi}_0^{\text{cl}}\\
\widetilde{\xi}_a^{\text{cl}}\\
\end{pmatrix}=\begin{pmatrix}
\xi^0\\
\xi^a+\xi^0v^a\\
\widetilde{\xi}_0^{\text{cl}}+\frac{1}{6}k_{abc}v^av^bv^c\xi^0 + \frac{1}{2}k_{abc}v^av^b\xi^c -\widetilde{\xi}_a^{\text{cl}}v^a\\
\widetilde{\xi}_a^{\text{cl}} -\frac{1}{2}\xi^0k_{abc}v^bv^c-k_{abc}v^b\xi^c\\
-\frac{1}{2}(\alpha^{\text{cl}}-\xi^i\widetilde{\xi}_i^{\text{cl}})-\frac{1}{6}(\xi^0)^2k_{abc}v^av^bv^c -\frac{1}{2}k_{abc}v^av^b\xi^c\xi^0  -\frac{1}{2}k_{abc}v^b\xi^a\xi^c\\
\end{pmatrix}
\end{equation}

where we have used $-\frac{1}{2}(\alpha^{\text{cl}}-\xi^i,\widetilde{\xi}_i^{\text{cl}})$ instead of $\alpha^{\text{cl}}$ because we want to relate to the Darboux coordinates (<ref>). On the other hand, using (<ref>) and (<ref>) one finds that under the action of $\mathbb{Z}^n$ the following holds

\begin{equation}
\widetilde{\xi}_0^{\text{inst}}\\
\widetilde{\xi}_a^{\text{inst}}\\
\alpha^{\text{inst}}\\
\end{pmatrix}=\begin{pmatrix}
\widetilde{\xi}_0^{\text{inst}} -\widetilde{\xi}_a^{\text{inst}}v^a\\
\widetilde{\xi}_a^{\text{inst}} \\
\alpha^{\text{inst}}\\
\end{pmatrix}\,.
\end{equation}

Hence, joining everything together one finds that  $(\xi^i,\widetilde{\xi}_i,\alpha)$ from (<ref>) transforms under the $\mathbb{Z}^{n}$ action by

\begin{equation}\label{vatwistor}
\xi^0\\
\xi^a\\
\widetilde{\xi}_0\\
\widetilde{\xi}_a\\
\alpha\\
\end{pmatrix}=\begin{pmatrix}
\xi^0\\
\xi^a+\xi^0v^a\\
\widetilde{\xi}_0+\frac{1}{6}k_{abc}v^av^bv^c\xi^0 + \frac{1}{2}k_{abc}v^av^b\xi^c -\widetilde{\xi}_av^a\\
\widetilde{\xi}_a -\frac{1}{2}\xi^0k_{abc}v^bv^c-k_{abc}v^b\xi^c\\
\alpha-\frac{1}{6}(\xi^0)^2k_{abc}v^av^bv^c -\frac{1}{2}k_{abc}v^av^b\xi^c\xi^0  -\frac{1}{2}k_{abc}v^b\xi^a\xi^c\,.
\end{pmatrix}
\end{equation}

In particular, the same holds for the Darboux coordinates $(\xi^i,\widetilde{\xi}_{i},\alpha_{c_{\ell}})$ obtained in Corollary <ref>, since $t$ is invariant under the lift of the $\mathbb{Z}^n$ action.  From the transformation rules (<ref>) it follows that

\begin{equation}
\D\alpha_{c_{\ell}} -\widetilde{\xi}_i\D \xi^i
\end{equation}

is invariant under the action of $\mathbb{Z}^n$ and acts holomorphically on a dense open subset of $\mathcal{Z}$ (here we use that Darboux coordinates for a holomorphic contact structure must be holomorphic coordinates). Since the action is globally defined and smooth, it follows that $\mathbb{Z}^n$ must act holomorphically globally on $\mathcal{Z}$ and preserve the contact structure.  The real structure is also trivially preserved, since the action on the fibers is trivial. Hence, we conclude that $\mathbb{Z}^n$ acts by twistor space automorphisms, and hence by isometries on $(\widetilde{N},g_{\overline{N}})$.

Finally, the case where $n_{\hat{\gamma}}=0$ for all $\hat{\gamma}\in \Lambda^{+}$ follows by a simplified version of the previous argument. The restriction of translations to $\mathbb{Z}^n$ is no longer required since $\mathfrak{F}^{\text{w.s}}$ does not have polylogarithm terms in this case, and the instanton corrections due to $\Omega(\gamma)$ only depends on the $\tau_2$ and $\zeta^0=\tau_1$ variables, which are left invariant under the full $\mathbb{R}^n$-action (<ref>).

Before joining the previous results and the ones from Section <ref> together, we will need the following lemma:

The group $\mathrm{SL}(2,\mathbb{R})\ltimes (\mathbb{R}^n\ltimes H_{2n+2})$ combining the S-duality action with the action of $\mathbb{R}^n\ltimes H_{2n+2}$ on $\overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}$ has $\mathrm{SL}(2,\mathbb{Z})\ltimes (\mathbb{Z}^n\ltimes H_{2n+2,D})$ as a subgroup.

The fact that the $\mathrm{SL}(2,\mathbb{R})$ S-duality action and the $\mathbb{R}^n\ltimes H_{2n+2}$ action combine into an action of a group of the form $\mathrm{SL}(2,\mathbb{R})\ltimes (\mathbb{R}^n\ltimes H_{2n+2})$ follows from <cit.>. On the other hand, to check that

\begin{equation}
\mathrm{SL}(2,\mathbb{Z})\ltimes (\mathbb{Z}^n\ltimes H_{2n+2,D})\subset \mathrm{SL}(2,\mathbb{R})\ltimes (\mathbb{R}^n\ltimes H_{2n+2})
\end{equation}

defines a subgroup, it is enough to check that the action by automorphisms of $\mathbb{Z}^n$ on $H_{2n+2}$ preserves the integrality contraint of $H_{2n+2,D}\subset H_{2n+2}$, and that the action by automorphisms of $\mathrm{SL}(2,\mathbb{Z})$ on $\mathbb{R}^n\ltimes H_{2n+2}$ preserves the integrality constraint of $\mathbb{Z}^n\ltimes H_{2n+2,D}\subset \mathbb{R}^n\ltimes H_{2n+2}$. The fact that the action by automorphisms of $\mathbb{Z}^n$ preserves the integrality constraint follows immediately from (<ref>). On the other hand, since $\mathrm{SL}(2,\mathbb{Z})$ is generated by $T$ and $S$ given in (<ref>), it is enough to check that the induced automorphism $\varphi_T, \varphi_S\in \text{Aut}(\mathbb{R}^n\ltimes H_{2n+2})$ preserve the integrality constraint of $\mathbb{Z}^n\ltimes H_{2n+2,D}$. From the fact that in the group $\mathrm{SL}(2,\mathbb{R})\ltimes (\mathbb{R}^n\ltimes H_{2n+2})$ we have that

\begin{equation}
(A,0,0)\cdot (1,v^a,(\eta^a,\eta_i,\kappa))\cdot(A^{-1},0,0)=(1,\varphi_A(v^a,(\eta^a,\eta_i,\kappa)))
\end{equation}

for $A\in \mathrm{SL}(2,\mathbb{R}), \; (v^a)\in \mathbb{R}^n, \; (\eta_a,\eta_i,\kappa)\in H_{2n+2}$, it is straightforward to use the actions of $\mathrm{SL}(2,\mathbb{R})$ and $\mathbb{R}\ltimes H_{2n+2}$ to compute that

\begin{equation}
\varphi_T(v^a,(\eta^a,\eta_i,\kappa))=(v^a,(\eta^a-v^a,...)), \quad \varphi_S(v^a,(\eta^a,\eta_i,\kappa))=(-\eta^a,(v^a,...))\,.
\end{equation}

It then follows that the integrality constraints of $\mathbb{Z}^n\ltimes H_{2n+2,D}$ are preserved by the automorphisms of $\mathrm{SL}(2,\mathbb{Z})$, and hence we obtain the desired result.

We can now state our main theorem:

Consider an instanton corrected q-map space $(\widetilde{N},g_{\overline{N}})$ of dimension $4n+4$, where are before, we take $\widetilde{N}$ to be the maximal domain of definition of $g_{\overline{N}}$. Let $T,S\in \mathrm{SL}(2,\mathbb{Z})$ be as in (<ref>). Then:

* The group

\begin{equation}\label{breakiso1}
\langle T \rangle \ltimes (\mathbb{Z}^n \ltimes H_{2n+2,D})
\end{equation}

acts by isometries on $(\widetilde{N},g_{\overline{N}})$,
where $\langle T \rangle \cong \mathbb{Z}$ is the subgroup generated by $T$.

* Assume that we take the one-loop parameter to be $c_{\ell}=\frac{\chi}{192\pi}$. We can always find a non-empty open subset $\widetilde{N}_S\subset \widetilde{N}$ where $(\widetilde{N}_S,g_{\overline{N}})$ carries an isometry group the the form

\begin{equation}
\langle S \rangle \ltimes (\mathbb{Z}^n \ltimes H_{2n+2,D}),
\end{equation}

where $\langle S \rangle \cong \mathbb{Z}/4\mathbb{Z}$ is the group generated by $S\in \mathrm{SL}(2,\mathbb{Z})$. Furthermore, if $\widetilde{N}_{\mathrm{SL}(2,\mathbb{Z})}\subset \widetilde{N}$ is  $\mathrm{SL}(2,\mathbb{Z})$-invariant,  then $\mathrm{SL}(2,\mathbb{Z})$ acts by isometries on $(\widetilde{N}_{\mathrm{SL}(2,\mathbb{Z})},g_{\overline{N}})$. In particular, if $\widetilde{N}$ is already invariant under $\mathrm{SL}(2,\mathbb{Z})$ then (<ref>) can be enhanced to

\begin{equation}
\mathrm{SL}(2,\mathbb{Z})\ltimes (\mathbb{Z}^n\ltimes H_{2n+2,D})\,.
\end{equation}

* Finally, if $n_{\hat{\gamma}}=0$ for all $\hat{\gamma}\in \Lambda^{+}$, then in the previous statements we can replace $\mathbb{Z}^n$ and $H_{2n+2,D}$ by $\mathbb{R}^n$ and $H_{2n+2}$, respectively. If furthermore we take  $\chi=c_{\ell}=0$ and $n_{\hat{\gamma}}=0$ for all $\hat{\gamma}\in \Lambda^{+}$, then we return to the tree-level q-map space case, where there is a connected $3n+6$ dimensional Lie group $G$ acting by isometries on $(\widetilde{N},g_{\overline{N}})$, see <cit.>. The group $G$ in particular contains the S-duality action by $\mathrm{SL}(2,\mathbb{R})$, an action by $\mathbb{R}^{n}\ltimes H_{2n+2}$, and a dilation action by $\mathbb{R}_{>0}$.  (<ref>).

The first statement of the theorem then follows from Lemma <ref>, Proposition <ref>,  Proposition <ref>, and the fact that the action of $T$ and $H_{2n+2,D}$ generate the action of $\text{Heis}_{2n+3,D}$. To check the second statement, notice that by Theorem <ref>, there is a non-empty open S-invariant subset $\widetilde{N}_S\subset \widetilde{N}$ where $S$ acts by isometries on $(\widetilde{N}_S,g_{\overline{N}})$. This subset is characterized in Theorem <ref> by

\begin{equation}\label{breakiso2}
\widetilde{N}_S=\{(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)\in \widetilde{N}\; | \; \epsilon<\tau_2, \;\; \epsilon< \frac{\tau_2}{|\tau_1|^2+|\tau_2^2|},\;\; t^a>K, \;\; |\tau|t^a>K\}\,.
\end{equation}

In particular, using that the action of $\mathbb{Z}^n$ and $H_{2n+2,D}$ leave $\tau_1$, $\tau_2$ and $t^a$ invariant, it follows that $\widetilde{N}_S$ carries an action by both groups. By the same proof as in Proposition <ref> and Proposition <ref>, it follows that $\mathbb{Z}^n$ and $H_{2n+2,D}$ must act by isometries on $(\widetilde{N}_S,g_{\overline{N}})$. The result then follows from Lemma <ref>. The last part of the second point and the final statement follow from Theorem <ref> and the aforementioned propositions and lemma.

§ AN EXAMPLE OF FULL S-DUALITY

As stated in Theorem <ref>, in order to guarantee that S-duality acts by isometries on the instanton corrected q-map metric, one needs to guarantee that the domain of the metric carries an action by the S-duality $\mathrm{SL}(2,\mathbb{Z})$. Here we explicitly give an example where the full  $\mathrm{SL}(2,\mathbb{Z})$ action by isometries can be guaranteed. We start by specifying the initial data as in Section <ref>.

* The PSR manifold $(\mathcal{H},g_{\mathcal{H}})$ is specified by the cubic polynomial $h:\mathbb{R}\to \mathbb{R}$ given by

\begin{equation}
\end{equation}

In particular $\mathcal{H}$ just reduces to a point. The corresponding PSK manifold $(\overline{M}^{\text{cl}},g_{\overline{M}^{\text{cl}}})$ obtained via the r-map has domain

\begin{equation}
\overline{M}^{\text{cl}}=\mathbb{R}+\I \mathbb{R}_{>0}\mathcal{H}=\mathbb{R} +\I\mathbb{R}_{>0}\,,
\end{equation}

with the corresponding CASK domain $(M^{\text{cl}},\mathfrak{F}^{\text{cl}})$ given by

\begin{equation}
M^{\text{cl}}=\{(Z^0,Z^1)=Z^0(1,z^1) \in \mathbb{C}^{2} \; | \; Z^0\in \mathbb{C}^{\times}, \; z^1 \in \overline{M}^{\text{cl}}\}, \quad \mathfrak{F}^{\text{cl}}=-\frac{1}{6}\frac{(Z^1)^3}{Z^0}\,.
\end{equation}

Furthermore we have $\Lambda^{+}=\text{span}_{\mathbb{Z}_{>0}}\{\gamma^1\}$ with the prepotential $\mathfrak{F}$ given by

\begin{equation}\label{prepotentialex}
\mathfrak{F}=-\frac{1}{6}\frac{(Z^1)^3}{Z^0}+\chi\frac{(Z^0)^2\zeta(3)}{2(2\pi \I )^3}\,, \quad \chi\in \mathbb{Z}_{>0}.
\end{equation}

Notice that we are restricting $\chi$ to be positive and we take $n_{\hat{\gamma}}=0$ for all $\hat{\gamma}\in \Lambda^{+}$.

* Since $n_{\hat{\gamma}}=0$ for all $\hat{\gamma}\in \Lambda^{+}$, we find that $M^{\text{cl}}=M^{q}$, where $M^{q}$ was defined in (<ref>). $M$ is then the maximal open subset of $M^{\text{cl}}$ where we have that $\text{Im}(\tau)$ has signature $(1,1)$ and $\text{Im}(\tau_{ij})Z^i\overline{Z}^j<0$. For our simple example, we can describe $M$ explicitly. Setting $z^1=b+\mathrm{i}t$, we have
\begin{equation}
\text{Im}(\tau_{00})=\frac{t^3}{3}-b^2t+\frac{\chi \zeta(3)}{(2\pi)^3}, \quad \text{Im}(\tau_{01})=bt, \quad
\text{Im}(\tau_{11})=-t,\,
\end{equation}

so that

\begin{equation}
\text{Det}(\text{Im}(\tau))=-\left(\frac{t^4}{3}+\frac{t\chi\zeta(3)}{(2\pi)^3}\right), \quad \text{Im}(\tau_{ij})Z^i\overline{Z}^j=|Z^0|^2\left(-\frac{2t^3}{3}+\frac{\chi\zeta(3)}{(2\pi)^3}\right)
\end{equation}

Since $\chi>0$ and $\text{Im}(\tau)$ is a $2\times 2$ matrix, $M$ is then given by

\begin{equation}\label{Mex}
M=\{(Z^0,Z^1)=Z^0(1,z^1)\in \mathbb{C}^2 \; | \; Z^0\in \mathbb{C}^{\times}, \;\; z^1\in \overline{M}^{\text{cl}}, \;\; \frac{t^3}{3}> \frac{\chi\zeta(3)}{2(2\pi)^3}\}\,.
\end{equation}

The prepotential $\mathfrak F$ induces on $M$ the structure of a CASK manifold
fibering over the complete PSK manifold $\overline M =\{ z^1\in \overline{M}^{\text{cl}} \mid \frac{t^3}{3}> \frac{\chi\zeta(3)}{2(2\pi)^3}\}$ with the Kähler potential $-\log (h(t) - \frac{\chi\zeta (3)}{4(2\pi )^3})$, as follows from the general results of [19]. The completeness
follows from <cit.> due to our assumption $\chi>0$. For $\chi<0$ the PSK metric is incomplete <cit.>.

* Over $M$, we consider  the trivial local system $\Gamma\to M$ with global sections $(\gamma^0,\gamma^1,\widetilde{\gamma}_0,\widetilde{\gamma}_1)$ defined by the CASK structure (recall Section <ref>), together with the canonical central charge $Z:M\to \Gamma^*\otimes \mathbb{C}$ given by $Z_{\gamma^i}=Z^i$, $Z_{\widetilde{\gamma}_i}=-\frac{\partial{\mathfrak{F}}}{\partial Z^i}$ for $i=0,1$. The BPS indices are given by

\begin{equation}\label{bpsindexex}
\begin{cases}
\Omega(q_0\gamma^0)=-\chi, \quad q_0\neq 0 \\
\Omega(\gamma)=0 \quad \text{else},\\
\end{cases}
\end{equation}

which has the required structure from Section <ref>.

We would like to now study the tensor $T$ from (<ref>) determining the domain of definition $N$ of the instanton corrected HK structure, as well as the functions $f$, $f_3$ and $g_{N}(V,V)$ from Section <ref>, determining the domain of the instanton corrected QK metric, as well as its signature.

Recalling that $Z^1/Z^0=z^1=b+\mathrm{i}t$, $|Z^0|=\tau_2$, $\zeta^0=\tau_1$, and $\tau=\tau_1+\mathrm{i}\tau_2$, we consider the open set $N\subset M^{\text{cl}}\times \mathbb{R}^{4}$ defined by

\begin{equation}
N:=\{(Z^0,Z^1,\zeta^0,\zeta^1,\widetilde{\zeta}_0,\widetilde{\zeta}_1)\in M^{\text{cl}}\times \mathbb{R}^4 \; | \; R_1(t,\tau)>0,\;\; R_2(t,\tau)>0\;\}
\end{equation}

\begin{equation}
\begin{split}
R_1(t,\tau)&:=\frac{t^3}{3}-\frac{\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{1}{|m\tau+n|^3}\\
R_2(t,\tau)&:=\frac{t^3}{3} -\frac{3\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{1}{|m\tau+n|^3}-\frac{(3\chi)^2}{(4\pi)^6}\left(\sum_{(m,n)\in \mathbb{Z}^2}\frac{1}{|m\tau+n|^3}\right)^2\left(R_1\right)^{-1}\,.
\end{split}
\end{equation}

The origin of these expressions will become clear with the following results. They will guarantee that the domain of the resulting QK metric carries an action of the S-duality $\mathrm{SL}(2,\mathbb{Z})$ and that the metric is positive definite.
Furthermore, we remark that $N$ is non-empty, since for a fixed $\tau$ the inequalities can be clearly achieved for $t$ sufficiently big. Furthermore, we remark that $R_1(t,\tau)>0$ already implies that $(Z^0,Z^1)\in M$, since (using that $\chi>0$)

\begin{equation}
\frac{\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{1}{|m\tau+n|^3}>\frac{\chi}{4(2\pi)^3}\sum_{n\in \mathbb{Z}-\{0\}}\frac{1}{|n|^3}=\frac{\chi\zeta(3)}{2(2\pi)^3}\;.
\end{equation}

The instanton corrected HK metric associated to the previous data as in Section <ref> is well defined on $N$ and of signature $(4,4)$.

The tensor determining the domain of definition of the HK geometry given in (<ref>) (compare (<ref>)) reduces in our case to:

\begin{equation}
T=-\text{Im}(\tau_{ij})\mathrm{d}Z^i\mathrm{d}\overline{Z}^j+\frac{\chi}{2\pi}\sum_{q_0\in \mathbb{Z}-\{0\}}\sum_{n>0}e^{-2\pi \mathrm{i}nq_0\zeta^0}q_0^2K_0(2\pi n\tau_2|q_0|)|\mathrm{d}Z^0|^2\,.
\end{equation}

We see that only the $T_{0\overline{0}}$ component of this tensor receives corrections due to BPS indices, while $T_{0\overline{1}}=T_{1\overline{0}}=-\text{Im}(\tau_{01})$ and $T_{1\overline{1}}=-\text{Im}(\tau_{11})$. The key thing is that we can Poisson resum $T_{0\overline{0}}$ using Lemma <ref> as follows

\begin{equation}
\begin{split}
&=-\frac{t^3}{3}+b^2t-\frac{\chi}{2(2\pi)^3}\sum_{n\neq 0}\frac{1}{n^2|n|}-\frac{\chi}{4(2\pi)^3}\partial_{\zeta^0}^2\mathcal{I}_0^{(2)}\\
&=-\frac{t^3}{3}+b^2t -\frac{\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\left(\frac{3(m\tau_1+n)^2}{|m\tau+n|^5}-\frac{1}{|m\tau+n|^3}\right)\,.
\end{split}
\end{equation}

From this computation, we find that the determinant along the horizontal directions gives

\begin{equation}
\text{Det}(T)=T_{0\overline{0}}T_{1\overline{1}}-(T_{0\overline{1}})^2=-\frac{t^4}{3} -\frac{t\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\left(\frac{3(m\tau_1+n)^2}{|m\tau+n|^5}-\frac{1}{|m\tau+n|^3}\right)\,.
\end{equation}

Since $\chi>0$ and $t>0$, we therefore have

\begin{equation}
\frac{t\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{3(m\tau_1+n)^2}{|m\tau+n|^5}>0
\end{equation}

and hence on the points of $N$ we have

\begin{equation}\label{detin}
\text{Det}(T)(p)<-\frac{t^4}{3}+\frac{t\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{1}{|m\tau+n|^3}=-t\cdot R_1(t,\tau)<0\,.
\end{equation}

This shows that the tensor $T$ is horizontally non-degenerate on $N$, and hence by Theorem <ref> we have that the instanton corrected HK metric $g_{N}$ is well defined on $N$. On the other hand by <cit.>, if the signature of the real matrix $(T_{i\overline{j}})$ is $(n,m)$ (called $M_{ij}$ in [25]), then the signature of $g_{N}$ is $(4n,4m)$. In our case $(T_{i\overline{j}})$ is a $2\times 2$ matrix, and since $\text{Det}(T)<0$, we must have that $(T_{i\overline{j}})$ has signature $(1,1)$. It then follows that the signature of $g_{N}$ is $(4,4)$.

If $f$, $f_3$ and $g_N(V,V)$ are the functions on $N$ defined on Section <ref>, and the $1$-loop parameter is taken to be $c_{\ell}=\frac{\chi}{192\pi}$, then on $N$ we have

\begin{equation}
f>0, \quad f_3<0, \quad g_N(V,V)>0\,.
\end{equation}

Using (<ref>) we obtain the following expressions for $f$ in our case:

\begin{equation}\label{fin}
\begin{split}
f&=2\pi\tau_2^2\left(\frac{2t^3}{3}\right)-\frac{\tau_2^2\chi}{2(2\pi)^2}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\frac{1}{|m\tau+n|^3}\\
&=4\pi \tau_2^2\left(\frac{t^3}{3}-\frac{\chi}{8(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{1}{|m\tau+n|^3}\right)\,\\
\end{split}
\end{equation}

So it follows that $f>0$ on $N$.

On the other hand, we remark that if we show that $f_3<0$ on $N$, then $g_N(V,V)>0$ follows due to the relation $g_N(V,V)=2(f-f_3)$ and the fact that we showed that $f>0$ on $N$.

To study $f_3$ we need to study the expression $g_{N}(V,V)$. Using <cit.> adapted to our conventions gives the following expression for $g_{N}$:

\begin{equation}
g_N=2\pi \left(T_{i\overline{j}}\mathrm{d}Z^i\mathrm{d}\overline{Z}^j +(W_i+W_i^{\text{inst}})T^{i\overline{j}}(\overline{W}_j+\overline{W}_j^{\text{inst}})\right),
\end{equation}

where (using the notation $\gamma=q_i(\gamma)\gamma^i$)

\begin{equation}
W_i=\mathrm{d}\zeta_{\widetilde{\gamma}_i}+\tau_{ij}\mathrm{d}\zeta_{\gamma^j}, \quad W_i^{\text{inst}}=-\sum_{\gamma}\Omega(\gamma)q_i(\gamma)\left(A_{\gamma}^{\text{inst}}-\mathrm{i}V_{\gamma}^{\text{inst}}\mathrm{d}\zeta_{\gamma}\right)\,.
\end{equation}

Using that $V=2\mathrm{i}(Z^i\partial_{Z^i}-\overline{Z}^i\partial{\overline{Z}^i})$ we can compute the evaluation of $g_{N}(V,V)$, which in our case gives

\begin{equation}\label{gnvv}
\begin{split}
g_{N}(V,V)=&8\pi T_{i\overline{j}}Z^i\overline{Z}^j+2\pi|W_0^{\text{inst}}(V)|^2T^{0\overline{0}}\,\\
&=8\pi \tau_2^2\left(\frac{2t^3}{3} -\frac{\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\left(\frac{3(m\tau_1+n)^2}{|m\tau+n|^5}-\frac{1}{|m\tau+n|^3}\right)\right)+2\pi|W_0^{\text{inst}}(V)|^2T^{0\overline{0}}
\end{split}
\end{equation}

where (using (<ref>) and Lemma <ref> for the expression of $W_0^{\text{inst}}$)

\begin{equation}
\begin{split}
W_0^{\text{inst}}(V)&=-\frac{\mathrm{i}\chi}{\pi}\tau_2\sum_{q_0\neq 0}\sum_{n>0}e^{2\pi \mathrm{i}n\zeta_{q_0\gamma^0}}q_0|q_0|K_1(2\pi n \tau_2|q_0|)\\
&=-\frac{3\chi \tau_2^2}{2(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2}\frac{(m\tau_1+n)m}{|m\tau+n|^5}\,.
\end{split}
\end{equation}

In the following, it will be convenient to find a lower bound for the negative term $|W_0^{\text{inst}}(V)|^2T^{0\overline{0}}$. We have

\begin{align*}
2\pi|W_0^{\text{inst}}(V)|^2T^{0\overline{0}}&=2\pi\left|\frac{3\chi \tau_2^2}{2(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2}\frac{(m\tau_1+n)m}{|m\tau+n|^5}\right|^2\frac{t}{\text{Det}(T)}\\
&>2\pi\left(\frac{3\chi \tau_2}{2(2\pi)^3}\right)^2\left|\sum_{(m,n)\in \mathbb{Z}^2}\frac{(m\tau_1+n)m\tau_2}{|m\tau+n|^5}\right|^2\left(-R_1(t,\tau)\right)^{-1}\\
&>2\pi\left(\frac{3\chi \tau_2}{2(2\pi)^3}\right)^2\left(\sum_{(m,n)\in \mathbb{Z}^2}\frac{|m\tau_1+n||m|\tau_2}{|m\tau+n|^5}\right)^2\left(-R_1(t,\tau)\right)^{-1}\\
&>2\pi\left(\frac{3\chi \tau_2}{2(2\pi)^3}\right)^2\left(\sum_{(m,n)\in \mathbb{Z}^2}\frac{(m\tau_1+n)^2+(m\tau_2)^2}{2|m\tau+n|^5}\right)^2\left(-R_1(t,\tau)\right)^{-1}\\
&=16\pi\tau_2^2\frac{(3\chi)^2}{2(4\pi)^6}\left(\sum_{(m,n)\in \mathbb{Z}^2}\frac{1}{|m\tau+n|^3}\right)^2\left(-R_1(t,\tau)\right)^{-1}\\ \numberthis \label{usefullowerbound}
\end{align*}

where in the first inequality we have used that $\text{Det}(T)<-t\cdot R_1<0$; in the second we used the triangle inequality, and in the last one we have used that $\sqrt{xy}\leq (x+y)/2$ for $x=(m\tau_1+n)^2$ and $y=(m\tau_2)^2$.

Using (<ref>) and (<ref>), we then find that $f_3=f-\frac{1}{2}g_N(V,V)$ gives

\begin{align*}\label{f3in}
f_3&=-4\pi \tau_2^2\left(\frac{t^3}{3} -\frac{\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\left(\frac{3(m\tau_1+n)^2}{|m\tau+n|^5}-\frac{3}{2|m\tau+n|^3}\right)\right)-\pi|W_0^{\text{inst}}(V)|^2T^{0\overline{0}}\\
&<-4\pi \tau_2^2\left(\frac{t^3}{3} -\frac{3\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{(m\tau_1+n)^2}{|m\tau+n|^5}\right)-\pi|W_0^{\text{inst}}(V)|^2T^{0\overline{0}}\\
&<-4\pi \tau_2^2\left(\frac{t^3}{3} -\frac{3\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{1}{|m\tau+n|^3}\right)-\pi|W_0^{\text{inst}}(V)|^2T^{0\overline{0}}\\
&<-4\pi \tau_2^2\left(\frac{t^3}{3} -\frac{3\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{1}{|m\tau+n|^3}\right)+8\pi\tau_2^2\frac{(3\chi)^2}{2^7(2\pi)^6}\left(\sum_{(m,n)\in \mathbb{Z}^2}\frac{1}{|m\tau+n|^3}\right)^2\left(R_1(t,\tau)\right)^{-1}\\
&=-4\pi\tau_2^2 R_2(t,\tau)<0\,. \numberthis
\end{align*}

The required inequalities therefore hold on $N$.

We then obtain as a corollary of Propositions <ref>, <ref>, and Theorem <ref> the following:

The instanton corrected q-map metric $g_{\overline{N}}$ associated to the prepotential (<ref>), BPS indices (<ref>) and with $1$-loop constant $c_{\ell}=\frac{\chi}{192\pi}$ is defined and positive definite on

\begin{equation}\label{tildeNex}
\widetilde{N}:=\{(\tau_2,b+\mathrm{i}t,\tau_1,\zeta^1,\widetilde{\zeta}_0,\widetilde{\zeta}_1,\sigma)\in \mathbb{R}_{>0}\times \overline{M}^{\text{cl}}\times \mathbb{R}^4\times \mathbb{R} \; | \; R_1(t,\tau)>0,\quad R_2(t,\tau)>0 \; \}.
\end{equation}

Because of Propositions <ref> and <ref>, we can take $N'=N$, where $N'$ is defined on (<ref>). It then follows from Theorem <ref> that $g_{\overline{N}}$ from (<ref>) is defined and positive definite on $\widetilde{N}$ from (<ref>).

The most important thing about $\widetilde{N}$ is the following:

$\widetilde{N}$ carries an action of S-duality.

Since the constraints in (<ref>) are given only on $t$ and $\tau$, we focus only on this variables. Under the action of $S\in \mathrm{SL}(2,\mathbb{Z})$ we have

\begin{equation}
t\to |\tau|t, \quad \tau \to -1/\tau\,,
\end{equation}

while under the action of $T\in \mathrm{SL}(2,\mathbb{Z})$ we have

\begin{equation}
t \to t, \quad \tau \to \tau+1\,.
\end{equation}

From the definition of $R_1$ and $R_2$, it follows immediately that they are invariant under $T$, and under $S$ they satisfy

\begin{equation}
R_i(|\tau|t,-1/\tau)=|\tau|^3R_i(t,\tau), \quad i=1,2
\end{equation}

so we conclude that $\widetilde{N}$ is invariant under $T$ and $S$.
Since $S$ and $T$ generate $\mathrm{SL}(2,\mathbb{Z})$ it then follows that $\widetilde{N}$ is invariant under $S$-duality.

The positive definite QK metric $(\widetilde{N},g_{\overline{N}})$ from Corollary <ref> has an effective action by isometries by the group $\mathrm{SL}(2,\mathbb{Z})\ltimes (\mathbb{R}\ltimes H_4)$.

By Proposition <ref>, $\widetilde{N}$ carries an action by $\mathrm{SL}(2,\mathbb{Z})$. Furthermore, since the definition of $\widetilde{N}$ in (<ref>) only constrains $\tau$ and $t$, and the action of $\mathbb{R}\ltimes H_4$ given in Section <ref> acts trivially on $\tau$ and $t$, $\widetilde{N}$ must carry an action of the group $\mathbb{R}\ltimes H_{4}$. It then follows from the Theorem <ref> that $(\widetilde{N},g_{\overline{N}})$ carries and action by isometries by the groups $\mathrm{SL}(2,\mathbb{Z})$ and $\mathbb{R}\ltimes H_4$ and hence by $\mathrm{SL}(2,\mathbb{Z})\ltimes (\mathbb{R}\ltimes H_4)$. We remark that we get $\mathbb{R}\ltimes H_4$ instead of $\mathbb{Z}\ltimes H_{4,D}$ since in our case we have $n_{\hat{\gamma}}=0$ for all $\hat{\gamma}$.

* Notice that the QK metric from Corollary <ref> is incomplete. This follows from the fact that the functions $R_i(t,\tau)$ defining $\widetilde{N}$ in (<ref>) are obtained from strict inequalities involving the conditions (see (<ref>),  (<ref>), (<ref>))

\begin{equation}\label{collin}
\text{Det}(T)<0,\quad  f>0,\quad f_3<0\,.
\end{equation}
This means that $g_{\overline{N}}$ extends to a positive definite metric on a bigger manifold containing (<ref>), and hence cannot be complete. It might be that (<ref>) is preserved by the S-duality action, but (except for $f>0$) this is not obvious from the formulas and the $\mathrm{SL}(2,\mathbb{Z})$ action.

* We also note that one can find a smaller S-duality invariant open subset $\widetilde{N}'\subset \widetilde{N}$ defined by a single relation $R(t,\tau)>0$. Namely, if we set

\begin{equation}\label{singlerel}
R(t,\tau):=\frac{t^3}{3}-\frac{9\chi}{8(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{1}{|m\tau+n|^3}
\end{equation}

and define

\begin{equation}
s(\tau):=\frac{\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{1}{|m\tau+n|^3},\,
\end{equation}

then we find that $R(t,\tau)>0$ implies that

\begin{equation}
\end{equation}

and furthermore

\begin{equation}
\begin{split}
R_1(t,\tau)R_2(t,\tau)&=\left(\frac{t^3}{3}-3s(\tau)\right)R_1  -\frac{(3\chi)^2}{(4\pi)^6}\left(\sum_{(m,n)\in \mathbb{Z}^2}\frac{1}{|m\tau+n|^3}\right)^2\\
&>\left(\frac{t^3}{3}-3s(\tau)\right)^2-\frac{(3\chi)^2}{(4\pi)^6}\left(\sum_{(m,n)\in \mathbb{Z}^2}\frac{1}{|m\tau+n|^3}\right)^2\\
\end{split}
\end{equation}

It then follows that $R(t,\tau)>0$ also implies $R_2(t,\tau)>0$.

Let $(\widetilde{N},g_{\overline{N}})$ be as in Corollary <ref>. Then:

* There is a free and properly discontinuous action by isometries of a discrete group of the form $\mathrm{SL}(2,\mathbb{Z})'\ltimes \Lambda$, where $\Lambda \subset \mathbb{R}\ltimes H_4$ is a lattice,  $\mathrm{SL}(2,\mathbb{Z})' \subset \mathrm{SL}(2,\mathbb{Z})$ is a finite index subgroup and the QK manifold $(\widetilde{N}/(\mathrm{SL}(2,\mathbb{Z})'\ltimes \Lambda),g_{\overline{N}})$ has finite volume.

* Furthermore, there is a submanifold with boundary $\hat{N}\subset \widetilde{N}$ where $\mathrm{SL}(2,\mathbb{Z})'\ltimes \Lambda$ acts and the quotient $(\hat{N}/(\mathrm{SL}(2,\mathbb{Z})'\ltimes \Lambda),g_{\overline{N}})$ gives a complete QK manifold with boundary and of finite volume.

By using the same argument as in <cit.>, we can find a lattice $\Lambda\subset \mathbb{R}\ltimes H_{4}$ such that $\mathrm{SL}(2,\mathbb{Z})\ltimes \Lambda$ is a lattice of the group of isometries $\mathrm{SL}(2,\mathbb{Z})\ltimes (\mathbb{R}\ltimes H_{4})$ (recall Corollary <ref>).
On the other hand, by using the $\mathrm{SL}(2,\mathbb{Z})$ action by isometries, we can assume that $\tau$ lies in the usual fundamental domain $\mathcal{F}_{\mathbb{H}}$ of the upper half place $\mathbb{H}\subset \mathbb{C}$ given by

\begin{equation}    \mathcal{F}_{\mathbb{H}}:=\{\tau \in \mathbb{H} \; | \; |\tau|>1, \;\; |\tau_1|<1/2\}\cup\{\tau \in \mathbb{H} \; | \; |\tau|\geq 1, \;\; \tau_1=-\frac{1}{2}\}\cup \{\tau \in \mathbb{H} \; | \; |\tau|=1, \;\; -\frac{1}{2}<\tau_1\leq 0\}\,.
\end{equation}

If we define

\begin{equation}
F:=\{(\tau_2,b+\mathrm{i} t,\tau_1,\zeta^1,\widetilde{\zeta}_0,\widetilde{\zeta}_1,\sigma)\in \widetilde{N} \; | \; \tau \in \mathcal{F}_{\mathbb{H}}\}
\end{equation}

we can then think of the quotient $\widetilde{N}/(\mathrm{SL}(2,\mathbb{Z})\ltimes \Lambda)$ as

\begin{equation}
\widetilde{N}/(\mathrm{SL}(2,\mathbb{Z})\ltimes \Lambda)=F/\Lambda\,.
\end{equation}

Note that $\Lambda$ actually acts on $F$, since $F$ is defined by restrictions on $\tau$ and $t$ and the latter are left invariant by the action of $\mathbb{R}\ltimes H_4$.
On $F/\Lambda$ the coordinates $b$, $\tau_1$, $\zeta^1$, $\widetilde{\zeta}_0$, $\widetilde{\zeta}_1$ and $\sigma$ are periodic. Furthermore, $\tau_2$ is bounded below by some positive constant (in fact, by $\sqrt{3}/2>0$), and the same holds for $t$, since the relation $R_1(t,\tau)>0$ defining $\widetilde{N}$ implies

\begin{equation}\label{tbound}
\frac{t^3}{3}>\frac{\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\frac{1}{|m\tau+n|^3}>\frac{\chi}{4(2\pi)^3}\sum_{n\in \mathbb{Z}-\{0\}}\frac{1}{|n|^3}=\frac{\chi\zeta(3)}{2(2\pi)^3}>0\;.
\end{equation}

On the other hand, $F$ has a boundary where $R_1(t,\tau)=0$ or $R_2(t,\tau)=0$. Since we know that the metric admits an extension beyond this boundary (see Remark <ref>), it cannot cause $F/\Lambda$ to have infinite volume. Furthermore,  on the end of $F/\Lambda$, where $\tau_2,t\to \infty$, the quantum correction terms in $g_{\overline{N}}$ are dominated by the tree-level q-map metric $(\overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}},g^{\text{cl}})$ associated to the PSR manifold $\mathcal{H}=\{\text{point}\}$ (defined by $h(t)=t^3/6$). Namely, in terms of the components of the metrics $g_{\overline{N}}$ and $g^{\text{cl}}$ we have $g_{\overline{N},ij}\sim g_{ij}^{\text{cl}}$ on the end of $F/\Lambda$ [This can be checked by looking at the $t$ and $\tau_2$ dependence of the quantum corrections. In our example, we only have the analog of a perturbative world-sheet correction encoded in the $\chi$ term of $\mathfrak{F}$, the analog of the perturbative 1-loop correction encoded in $c_{\ell}=\chi/192\pi$, and the analog of D(-1) instanton corrections encoded in terms with $\Omega(q_0\gamma^0)=-\chi$. The D(-1) corrections are exponentially suppressed as $\tau_2\to \infty$, while the pertubative corrections can be safely ignored when $\tau_{2},t\to \infty$, since these are dominated by the classical terms of the tree-level q-map metric.]. We can therefore use the volume density of $g^{\text{cl}}$ to asymptotically approximate the volume density of $g_{\overline{N}}$ on this end. But by <cit.>, we know that $\overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}/(\mathrm{SL}(2,\mathbb{Z})\ltimes \Lambda)$ has finite volume with respect to $g_{\text{cl}}$, so that $\widetilde{N}/(\mathrm{SL}(2,\mathbb{Z})\ltimes \Lambda)\subset \overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}/(\mathrm{SL}(2,\mathbb{Z})\ltimes \Lambda)$ also has finite volume with respect to $g^{\text{cl}}$. It follows that the end of $F/\Lambda$ where $\tau_2,t\to \infty$ must give a finite volume contribution with respect to $g_{\overline{N}}$. Hence, $(\widetilde{N}/(\mathrm{SL}(2,\mathbb{Z})\ltimes \Lambda),g_{\overline{N}})$ has finite volume.

The action of $\mathrm{SL}(2,\mathbb{Z})\ltimes \Lambda$ on $\widetilde{N}$ is easily seen to be properly discontinuous, however it is not free, since the points satisfying $\tau_2=\mathrm{i}$, $b=\tau_1=c^1=c_0=\psi=0$ are fixed by the subgroup $\langle S \rangle=\mathbb{Z}/4\mathbb{Z}\subset \mathrm{SL}(2,\mathbb{Z})$. We can nevertheless pick a finite index subgroup $\mathrm{SL}(2,\mathbb{Z})'\subset \mathrm{SL}(2,\mathbb{Z})$ that acts freely by choosing a finite index subgroup that intersects $\langle S \rangle$ only at the identity (for example by picking the kernel of the homomorphism $\mathrm{SL}(2,\mathbb{Z})\to \mathrm{SL}(2,\mathbb{Z}/N\mathbb{Z})$ reducing the entries modulo $N$ for $N\geq 3$). We therefore get a free, properly discontinuous action of $\mathrm{SL}(2,\mathbb{Z})'\ltimes \Lambda$ on $\widetilde{N}$. Since $(\widetilde{N}/(\mathrm{SL}(2,\mathbb{Z})\ltimes \Lambda),g_{\overline{N}})$ has finite volume and $\mathrm{SL}(2,\mathbb{Z})'\subset \mathrm{SL}(2,\mathbb{Z})$ is a finite index subgroup, it follows that $(\widetilde{N}/(\mathrm{SL}(2,\mathbb{Z})'\ltimes \Lambda),g_{\overline{N}})$ is a QK manifold of finite volume.

We now prove the last statement. We define $\hat{N}$ by

\begin{equation}\label{hatNex}
\hat{N}:=\{(\tau_2,b+\mathrm{i} t,\tau_1,\zeta^1,\widetilde{\zeta}_0,\widetilde{\zeta}_1,\sigma)\in \widetilde{N} \; | \; R(t,\tau)\geq 0\; \},
\end{equation}

where $\widetilde{N}$ is as in Corollary <ref> and $R(t,\tau)$ was defined in (<ref>). It is easy to check that the function $R(t,\tau)$ is regular on the level set $R(t,\tau)=0$ (for example, by using that $t>0$ on $\widetilde{N}$, due to (<ref>)), so that $\hat{N}$ is a smooth manifold with boundary.

By the same argument of Proposition <ref> and Corollary <ref> it follows that the action of $\mathrm{SL}(2,\mathbb{Z})\ltimes (\mathbb{R}\ltimes H_4)$ on $\widetilde{N}$ restricts to $\hat{N}$. Furthermore, since $(\widetilde{N}/(\mathrm{SL}(2,\mathbb{Z})'\ltimes \Lambda),g_{\overline{N}})$ has finite volume,  $(\hat{N}/(\mathrm{SL}(2,\mathbb{Z})\ltimes \Lambda),g_{\overline{N}})$ also has finite volume. Finally, to show that $(\hat{N}/(\mathrm{SL}(2,\mathbb{Z})\ltimes \Lambda),g_{\overline{N}})$ is complete, recall that a Riemannian manifold with boundary is complete if it is a complete metric space with the induced distance function. This follows from the fact that it contains the boundary points satisfying $R(t,\tau)=0$; the coordinates $b$, $\tau_1$, $\zeta^1$, $\widetilde{\zeta}_0$, $\widetilde{\zeta}_1$, $\sigma$ are periodic; and the metric in the end where $\tau_2,t\to \infty$ is complete, since it can be approximated by the complete tree-level q-map metric $g^{\text{cl}}$ (the completeness of $g^{\text{cl}}$ follows from <cit.>).

We finish with the following remark about a related example associated to the resolved conifold.

To the resolved conifold one can associate a natural holomorphic prepotential of the required form (<ref>),

\begin{equation}
\mathfrak{F}=-\frac{1}{6}\frac{(Z^1)^3}{Z^0}+\chi\frac{(Z^0)^2\zeta(3)}{2(2\pi \I )^3}-\frac{(Z^0)^2}{(2\pi \I)^3}n_{\gamma^1}\mathrm{Li}_3(e^{2\pi \I Z^1/Z^0})\,, \quad \chi=2, \quad n_{\gamma^1}=1\,.
\end{equation}

This form of the prepotential can be motivated by considering a certain extension of the Picard-Fuchs operators, see [30]. On the other hand, the BPS spectrum of the resolved conifold also has the required form (<ref>), with

\begin{equation}
\begin{cases}
\Omega(q_0\gamma^0)=-\chi=-2, \quad q_0\in \mathbb{Z}-\{0\}\\
\Omega(q_0\gamma^0\pm \gamma^1)=\Omega(\pm \gamma^1)=n_{\gamma^1}=1 \quad \text{for $\gamma^1 \in \Lambda^+$, $q_0\in \mathbb{Z}$}\\
\Omega(\gamma)=0 \quad \text{else}.\\
\end{cases}
\end{equation}

We can therefore apply the construction of Section <ref> and obtain an instanton corrected q-map space.
We expect that a similar argument as in the example can be done in order to find an S-duality invariant open subset of the domain of definition of the associated instanton corrected q-map space. As in Theorem <ref>, we also expect that it admits a quotient of finite volume. Furthermore, we remark that since the resolved conifold is a non-compact Calabi-Yau 3-fold without compact divisors, the possible quantum corrections in the type IIB string theory language simplify to just the world-sheet corrections and the 1-loop correction $c_\ell$ together with the D(-1) and D1-instanton corrections. Since these are all accounted for in the above construction, one could expect that that the resulting instanton corrected q-map space with its maximal domain of definition might be complete. We leave the study of this question for future work.

§ INTEGRAL IDENTITIES IN TERMS OF BESSEL FUNCTIONS

In this appendix we will prove the following lemma, relating certain integrals with infinite sums of Bessel functions. We recall that given $\gamma=q_i\gamma^i$ and the functions $\xi^i$ from (<ref>), we use the notation from Section <ref>, where $\xi_{\gamma}=q_i\xi^i=q_i(\zeta^i-\mathrm{i}R(t^{-1}z^i+t\overline{z}^i))=-\zeta_{\gamma}-\mathrm{i}R(t^{-1}\widetilde{Z}_{\gamma}+t\overline{\widetilde{Z}_{\gamma}})$.

Consider the modified Bessel functions of the second kind $K_{\nu}:\mathbb{R}_{>0}\to \mathbb{R}$, which have the following integral representation

\begin{equation}\label{Besselintrep}
K_{\nu}(x)=\int_{0}^{\infty}dt \exp(-x\cosh(t))\cosh(\nu t), \quad \nu=0,1,2,...
\end{equation}

Following the notation from Section <ref>, we have the following identities for $\gamma=q_i\gamma^i$

\begin{equation}\label{Bessel1}
\int_{l_{\gamma}}\frac{d\zeta}{\zeta^{1-q}}\frac{\exp(2\pi\I \xi_{\gamma})}{1-\exp(2\pi\I \xi_{\gamma})}=
\begin{cases}
2\frac{|\widetilde{Z}_{\gamma}|^2}{\widetilde{Z}_{\gamma}^2}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}\left(K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)+\frac{K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)}{2\pi Rn|\widetilde{Z}_{\gamma}|}\right), \quad q=-2\\
-2\frac{|\widetilde{Z}_{\gamma}|}{\widetilde{Z}_{\gamma}}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_1(4\pi Rn|\widetilde{Z}_{\gamma}|), \quad q=-1\\
2\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|), \quad q=0\\
-2\frac{\widetilde{Z}_{\gamma}}{|\widetilde{Z}_{\gamma}|}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_1(4\pi Rn|\widetilde{Z}_{\gamma}|),\quad q=1\,\\
2\frac{\widetilde{Z}_{\gamma}^2}{|\widetilde{Z}_{\gamma}|^2}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}\left(K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)+\frac{K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)}{2\pi Rn|\widetilde{Z}_{\gamma}|}\right), \quad q=2\\
\end{cases}
\end{equation}

\begin{equation}\label{Bessel2}
\int_{l_{\gamma}}\frac{d\zeta}{\zeta^{1-q}}\log(1-\exp(2\pi\I \xi_{\gamma}))=\begin{cases}
2\frac{|\widetilde{Z}_{\gamma}|}{\widetilde{Z}_{\gamma}}\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}K_1(4\pi Rn|\widetilde{Z}_{\gamma}|), \quad q=-1\\
-2\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|), \quad q=0\\
2\frac{\widetilde{Z}_{\gamma}}{|\widetilde{Z}_{\gamma}|}\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}K_1(4\pi Rn|\widetilde{Z}_{\gamma}|),\quad q=1\,.
\end{cases}
\end{equation}

We start by computing the integrals of the form

\begin{equation}
\int_{l_{\gamma}}\frac{d\zeta}{\zeta^{1-q}}\frac{\exp(2\pi\I \xi_{\gamma})}{1-\exp(2\pi\I \xi_{\gamma})}, \quad q=-2,-1,0,1,2\,,
\end{equation}

\begin{equation}
l_{\gamma}=\{t \; | \; \widetilde{Z}_{\gamma}/t\in \mathbb{R}_{<0}\}\,.
\end{equation}

Setting $\zeta=-s\widetilde{Z}_{\gamma}/|\widetilde{Z}_{\gamma}|$ for $s\in (0,\infty)$, and using that $|\exp(2\pi\I \xi_{\gamma})|_{l_{\gamma}}|<1$, we have

\begin{align*}
\int_{l_{\gamma}}\frac{d\zeta}{\zeta^{1-q}}\frac{\exp(2\pi\I \xi_{\gamma})}{1-\exp(2\pi\I \xi_{\gamma})}&=(-1)^{q}\frac{\widetilde{Z}_{\gamma}^q}{|\widetilde{Z}_{\gamma}|^q}\int_{0}^{\infty}\frac{ds}{s^{1-q}}\frac{\exp(-2\pi R|\widetilde{Z}_{\gamma}|(s^{-1}+s)-2\pi\I \zeta_{\gamma})}{1-\exp(-2\pi R|\widetilde{Z}_{\gamma}|(s^{-1}+s)-2\pi\I \zeta_{\gamma})}\\
&=(-1)^{q}\frac{\widetilde{Z}_{\gamma}^q}{|\widetilde{Z}_{\gamma}|^q}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}\int_{0}^{\infty}\frac{ds}{s^{1-q}}\exp(-2\pi Rn|\widetilde{Z}_{\gamma}|(s^{-1}+s))\\
&=(-1)^{q}\frac{\widetilde{Z}_{\gamma}^q}{|\widetilde{Z}_{\gamma}|^q}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}\int_{-\infty}^{\infty}dxe^{qx}\exp(-4\pi Rn|\widetilde{Z}_{\gamma}|\cosh(x))\\
&=2(-1)^{q}\frac{\widetilde{Z}_{\gamma}^q}{|\widetilde{Z}_{\gamma}|^q}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}\int_{0}^{\infty}dx\exp(-4\pi Rn|\widetilde{Z}_{\gamma}|\cosh(x))\cosh(qx)\\
&=2(-1)^{q}\frac{\widetilde{Z}_{\gamma}^q}{|\widetilde{Z}_{\gamma}|^q}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}\int_{0}^{\infty}dx\exp(-4\pi Rn|\widetilde{Z}_{\gamma}|\cosh(x))\cosh(|q|x)\\
&=2(-1)^{q}\frac{\widetilde{Z}_{\gamma}^q}{|\widetilde{Z}_{\gamma}|^q}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_{|q|}(4\pi Rn|\widetilde{Z}_{\gamma}|)\numberthis\\
\end{align*}

Hence, we obtain (<ref>), where for the case $q=\pm 2$ we have used the identity $K_2(x)=K_0(x)+2K_1(x)/x$.

We can similarly compute integrals of the form

\begin{equation}
\int_{l_{\gamma}}\frac{d\zeta}{\zeta^{1-q}}\log(1-\exp(2\pi\I \xi_{\gamma})), \quad q=-1,0,1
\end{equation}

Indeed since $|\exp(\mathrm{2\pi\I }\xi_{\gamma})|_{l_{\gamma}}|<1$, we can expand $\log(1-x)=-\sum_{n>0}x^n/n$ and obtain

\begin{equation}
\begin{split}
\int_{l_{\gamma}}\frac{d\zeta}{\zeta^{1-q}}\log(1-\exp(2\pi\I \xi_{\gamma}))&=-\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}\int_{l_{\gamma}}\frac{d\zeta}{\zeta^{1-q}}\exp(2\pi nR\widetilde{Z}_{\gamma}/\zeta +nR\overline{Z}_{\gamma}\zeta)\\
&=(-1)^{q+1}\frac{\widetilde{Z}_{\gamma}^q}{|\widetilde{Z}_{\gamma}|^q}\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}\int_{0}^{\infty}\frac{ds}{s^{1-q}}\exp(-2\pi Rn|\widetilde{Z}_{\gamma}|(s^{-1}+s))\\
&=2(-1)^{q+1}\frac{\widetilde{Z}_{\gamma}^q}{|\widetilde{Z}_{\gamma}|^q}\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}K_{|q|}(4\pi Rn|\widetilde{Z}_{\gamma}|)\\
\end{split}
\end{equation}

so that we obtain (<ref>).

§ TYPE IIA DARBOUX COORDINATES FOR INSTANTON CORRECTED C-MAP SPACES

Here we give a proof of Theorem <ref>.

We note that $\widetilde{\xi}_i$ and $\alpha$ can be written as $\widetilde{\xi}_i=\widetilde{\xi}_i^{c}+\widetilde{\xi}^{\text{inst}}$ and $\alpha=\alpha^c+\alpha^{\text{inst}}$, where the index $c$ denotes the part of the coordinates that coincides with the c-map case of (<ref>), and $\text{inst}$ denotes the terms involving $\Omega(\gamma)$. We have a similar decomposition for $f$ and $\theta_i^P$ using (<ref>), and hence $\lambda$ by using (<ref>). Because of Proposition <ref> it is then enough to show that

\begin{equation}
f^{\text{inst}}\frac{\D t}{t}+t^{-1}\theta_+^{P,\text{inst}}|_{\overline{N}} -2\mathrm{i}\theta_3^{P,\text{inst}}|_{\overline{N}} +t\theta_-^{P,\text{inst}}|_{\overline{N}}=-2\pi \I \left(\D \alpha^{\text{inst}}+\widetilde{\xi}_i^{\text{inst}}\D \xi^i -\xi^i\D \widetilde{\xi}_i^{\text{inst}}\right)\,.
\end{equation}

We will check this by a direct computation.

First we compute $\D \alpha^{\text{inst}}$. From the expression of $\mathcal{W}$ in (<ref>) we compute $\D\mathcal{W}$:

\begin{align*}
\D \mathcal{W}=&R\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\widetilde{Z}_{\gamma}}{2(\rho+c_{\ell})}\D \rho+\frac{\widetilde{Z}_{\gamma}}{2}\D \mathcal{K} +\D \widetilde{Z}_{\gamma}\right)\\
&-2\pi\I R\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\D \zeta_{\gamma}\\
&-2\pi R^2\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D \rho}{\rho+c_{\ell}}+\D \mathcal{K}+\frac{\D \widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}+\frac{\D \overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}\right)\\\numberthis
\end{align*}

where we have used on the last line that $(K_0(x))'=-K_1(x)$.

On the other hand, we have

\begin{equation}
\begin{split}
\D &\left(-\frac{1}{2\pi}\sum_{\gamma}\Omega(\gamma)\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{t+\zeta}{t-\zeta}\mathrm{L} (\exp(2\pi\I \xi_{\gamma}))\right)\\
=&-\frac{1}{2\pi}\sum_{\gamma}\Omega(\gamma)\D t\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\partial_t\left(\frac{t+\zeta}{t-\zeta}\right)\mathrm{L} (\exp(2\pi\I \xi_\gamma(\zeta)))\\
&-\sum_{\gamma}\Omega(\gamma)\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{t+\zeta}{t-\zeta}\left(-\frac{\I }{2}\log(1-\exp(2\pi\I \xi_{\gamma})) +\frac{1}{2}\frac{\exp(\I \xi_{\gamma})}{1-\exp(2\pi\I \xi_{\gamma})}2\pi\xi_{\gamma}\right)\D \xi_{\gamma}(\zeta)
\end{split}
\end{equation}

We can then join everything together to conclude that

\begin{align*}
-2\pi \I \D \alpha^{\text{inst}}=&-\frac{1}{2\pi}\left(-t^{-2}\mathcal{W}+\overline{\mathcal{W}}\right)\D t\\
&-t^{-1}\frac{R}{2\pi}\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\widetilde{Z}_{\gamma}}{2(\rho+c_{\ell})}\D \rho+\frac{\widetilde{Z}_{\gamma}}{2}\D \mathcal{K} +\D \widetilde{Z}_{\gamma}\right)\\
&+t^{-1}\I R\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\D \zeta_{\gamma}\\
&+t^{-1}R^2\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D \rho}{\rho+c_{\ell}}+\D \mathcal{K}+\frac{\D \widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}+\frac{\D \overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}\right)\\
&+t\cdot\left(\text{complex conjugate of $t^{-1}$ term factor}\right)\\ &+\frac{1}{4\pi^2}\sum_{\gamma}\Omega(\gamma)\D t\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\partial_t\left(\frac{t+\zeta}{t-\zeta}\right)\mathrm{L} (\exp(2\pi\I \xi_\gamma(\zeta)))\\
&+\frac{1}{2\pi}\sum_{\gamma}\Omega(\gamma)\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{t+\zeta}{t-\zeta}\left(-\frac{\I }{2}\log(1-\exp(2\pi\I \xi_{\gamma})) +\frac{1}{2}\frac{\exp(2\pi\I \xi_{\gamma})}{1-\exp(2\pi\I \xi_{\gamma})}2\pi\xi_{\gamma}\right)\D \xi_{\gamma}(\zeta)\numberthis\label{conttypeiia1}
\end{align*}

On the other hand, for the terms $ \widetilde{\xi}_i^{\text{inst}}\D\xi^i- \xi^i\D\widetilde{\xi}_i^{\text{inst}}$ we have

\begin{equation}\label{conttypeiia2}
\begin{split}
-2\pi \I\left(\widetilde{\xi}_i^{\text{inst}}\D \xi^i- \xi^i\D \widetilde{\xi}_i^{\text{inst}}\right)=&-\frac{1}{4\pi \I }\sum_{\gamma}\Omega(\gamma)\D \xi_{\gamma}(t)\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{t+\zeta}{t-\zeta}\log(1-\exp(2\pi\I \xi_{\gamma}))\\
&+\frac{1}{4\pi \I }\sum_{\gamma}\Omega(\gamma)\xi_{\gamma}(t)\D t \int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\partial_t\left(\frac{t+\zeta}{t-\zeta}\right)\log(1-\exp(2\pi\I \xi_{\gamma}(\zeta)))\\
&-\frac{1}{2 }\sum_{\gamma}\Omega(\gamma)\xi_{\gamma}(t) \int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{t+\zeta}{t-\zeta}\frac{\exp(2\pi\I \xi_{\gamma}(\zeta))}{1-\exp(2\pi\I \xi_{\gamma}(\zeta))}\D \xi_{\gamma}(\zeta)\,.
\end{split}
\end{equation}

To start combining terms, we notice that

\begin{equation}
\frac{t+\zeta}{t-\zeta}(\D \xi_{\gamma}(t)-\D \xi_{\gamma}(\zeta))=\left(\frac{1}{\zeta}+\frac{1}{t}\right)\I \D (R\widetilde{Z}_{\gamma})-(t+\zeta)\I \D (R\overline{\widetilde{Z}}_{\gamma})-\frac{t+\zeta}{t-\zeta}\I R(-t^{-2}\widetilde{Z}_{\gamma} +\overline{\widetilde{Z}}_{\gamma})\D t
\end{equation}

\begin{equation}
\frac{t+\zeta}{t-\zeta}(\xi_{\gamma}(t)-\xi_{\gamma}(\zeta))=\left(\frac{1}{\zeta}+\frac{1}{t}\right)\I R\widetilde{Z}_{\gamma}-(t+\zeta)\I R\overline{\widetilde{Z}}_{\gamma}
\end{equation}

so that we can combine the first and third term of (<ref>) with the last term of (<ref>) into

\begin{align*}
&-\frac{1}{4\pi \I }\sum_{\gamma}\Omega(\gamma)\D \xi_{\gamma}(t)\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{t+\zeta}{t-\zeta}\log(1-\exp(2\pi\I \xi_{\gamma}))-\frac{1}{2 }\sum_{\gamma}\Omega(\gamma)\xi_{\gamma}(t) \int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{t+\zeta}{t-\zeta}\frac{\exp(2\pi\I \xi_{\gamma}(\zeta))}{1-\exp(2\pi\I \xi_{\gamma}(\zeta))}\D \xi_{\gamma}(\zeta)\\
&\quad +\frac{1}{2\pi}\sum_{\gamma}\Omega(\gamma)\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{t+\zeta}{t-\zeta}\left(-\frac{\I }{2}\log(1-\exp(2\pi\I \xi_{\gamma})) +\frac{1}{2}\frac{\exp(2\pi\I \xi_{\gamma})}{1-\exp(2\pi \I\xi_{\gamma})}2\pi\xi_{\gamma}(\zeta)\right)\D \xi_{\gamma}(\zeta)\\
&=-\frac{1}{4\pi \I }\sum_{\gamma}\Omega(\gamma)\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{t+\zeta}{t-\zeta}\log(1-\exp(2\pi\I \xi_{\gamma}))(\D \xi_{\gamma}(t)-\D \xi_{\gamma}(\zeta)) \\
&\quad -\frac{1}{2 }\sum_{\gamma}\Omega(\gamma) \int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{t+\zeta}{t-\zeta}\frac{\exp(2\pi\I \xi_{\gamma}(\zeta))}{1-\exp(2\pi\I \xi_{\gamma}(\zeta))}(\xi_{\gamma}(t)-\xi_{\gamma}(\zeta))\D \xi_{\gamma}(\zeta)\\
&=-\frac{1}{4\pi \I }\sum_{\gamma}\Omega(\gamma)\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\log(1-\exp(2\pi\I \xi_{\gamma}))\left(\left(\frac{1}{\zeta}+\frac{1}{t}\right)\I \D (R\widetilde{Z}_{\gamma})-(t+\zeta)\I \D (R\overline{\widetilde{Z}}_{\gamma})-\frac{t+\zeta}{t-\zeta}\I R(-t^{-2}\widetilde{Z}_{\gamma} +\overline{\widetilde{Z}}_{\gamma})\D t\right) \\
&\quad +\frac{1}{2 }\sum_{\gamma}\Omega(\gamma) \int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{\exp(2\pi\I \xi_{\gamma}(\zeta))}{1-\exp(2\pi\I \xi_{\gamma}(\zeta))}\left(\left(\frac{1}{\zeta}+\frac{1}{t}\right)\I R\widetilde{Z}_{\gamma}-(t+\zeta)\I R\overline{\widetilde{Z}}_{\gamma}\right)\left(\D \zeta_{\gamma} +\I \zeta^{-1}\D (R\widetilde{Z}_{\gamma}) +\I \zeta \D (R\overline{\widetilde{Z}}_{\gamma})\right)\,.\numberthis\\
\end{align*}

We can rewrite the previous integrals in the last equality in terms of Bessel functions by using the identities (<ref>) and (<ref>), obtaining

\begin{align*}
=&\frac{1}{4\pi }\sum_{\gamma}\Omega(\gamma)\D t\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\log(1-\exp(2\pi\I \xi_{\gamma}))\left(\frac{t+\zeta}{t-\zeta}R(-t^{-2}\widetilde{Z}_{\gamma} +\overline{\widetilde{Z}}_{\gamma})\right)\\
&- \frac{1}{2\pi}\sum_{\gamma}\Omega(\gamma)\D (R\widetilde{Z}_{\gamma})\left(\sum_{n>0}\frac{|\widetilde{Z}_{\gamma}|}{\widetilde{Z}_{\gamma}}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)-\frac{1}{t}\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\right)\\
&-\frac{1}{2\pi}\sum_{\gamma}\Omega(\gamma)\D (R\overline{\widetilde{Z}}_{\gamma})\left(-\sum_{n>0}\frac{\widetilde{Z}_{\gamma}}{|\widetilde{Z}_{\gamma}|}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)+t\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\right)\\
&-\sum_{\gamma}\Omega(\gamma)\frac{|\widetilde{Z}_{\gamma}|^2}{\widetilde{Z}_{\gamma}}R\D (R\widetilde{Z}_{\gamma})\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}\left(K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)+\frac{K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)}{2\pi Rn|\widetilde{Z}_{\gamma}|}\right)\\
&+\sum_{\gamma}\Omega(\gamma)\left(-\I \widetilde{Z}_{\gamma}R\D \zeta_{\gamma}+\left(\widetilde{Z}_{\gamma}/t-t\overline{\widetilde{Z}}_{\gamma}\right)R\D (R\widetilde{Z}_{\gamma})\right)\frac{|\widetilde{Z}_{\gamma}|}{\widetilde{Z}_{\gamma}}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\\
&-\sum_{\gamma}\Omega(\gamma)\left(-(\widetilde{Z}_{\gamma}/t-t\overline{\widetilde{Z}}_{\gamma})\I R\D \zeta_{\gamma} +R\widetilde{Z}_{\gamma}\D (R\overline{\widetilde{Z}}_{\gamma}) - \overline{\widetilde{Z}}_{\gamma}R\D (R\widetilde{Z}_{\gamma})\right)\sum_{n>0}e^{-2\pi \I n\zeta_{\gamma}}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\\
&+\sum_{\gamma}\Omega(\gamma)\left(\overline{\widetilde{Z}}_{\gamma}\I R\D \zeta_{\gamma}+(\widetilde{Z}_{\gamma}/t-t\overline{\widetilde{Z}}_{\gamma})R\D (R\overline{\widetilde{Z}}_{\gamma})\right)\frac{\widetilde{Z}_{\gamma}}{|\widetilde{Z}_{\gamma}|}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\\
&+\sum_{\gamma}\Omega(\gamma)\overline{\widetilde{Z}}_{\gamma}R\D (R\overline{\widetilde{Z}}_{\gamma})\frac{\widetilde{Z}_{\gamma}^2}{|\widetilde{Z}_{\gamma}|^2}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}\left(K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)+\frac{K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)}{2\pi Rn|\widetilde{Z}_{\gamma}|}\right) \numberthis
\end{align*}

Overall, we obtain the following $t^{-1}$ term for $-2\pi \I(\D \alpha^{\text{inst}}+\widetilde{\xi}_i^{\text{inst}}\D\xi^i-\xi^i\D \widetilde{\xi}_i^{\text{inst}})$:

\begin{align*}
&\frac{R}{2\pi}\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\widetilde{Z}_{\gamma}}{2(\rho+c_{\ell})}\D \rho+\frac{\widetilde{Z}_{\gamma}}{2}\D \mathcal{K} +\D \widetilde{Z}_{\gamma}\right)\\
&+R^2\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D \rho}{2(\rho+c_{\ell})}+\frac{\D \mathcal{K}}{2} +\frac{\D \widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}\right)\\
&+\I R\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\D \zeta_{\gamma}\\
&+R^2\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D \rho}{2(\rho+c_{\ell})}+\frac{\D \mathcal{K}}{2} +\frac{\D \overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}\right)\\
&-\frac{R}{2\pi}\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\widetilde{Z}_{\gamma}}{2(\rho+c_{\ell})}\D \rho+\frac{\widetilde{Z}_{\gamma}}{2}\D \mathcal{K} +\D \widetilde{Z}_{\gamma}\right)\\
&+\I R\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\D \zeta_{\gamma}\\
&+R^2\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D \rho}{\rho+c_{\ell}}+\D \mathcal{K}+\frac{\D \widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}+\frac{\D \overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}\right)\\
=&2\I R\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\D \zeta_{\gamma}\\
&+2R^2\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D \rho}{\rho+c_{\ell}}+\D \mathcal{K}+\frac{\D \widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}+\frac{\D \overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}\right)\\
\end{align*}

By comparing with (<ref>) we see that this matches $\theta_{+}^{P,\text{inst}}|_{\overline{N}}$. The $t$ term follows from the $t^{-1}$, by noticing that all the $t$-term are conjugates of the $t^{-1}$-term. In particular, the $t$-term will match $\theta_{-}^{\text{inst}}=\overline{\theta_{+}^{\text{inst}}}$.

Now we collect the $t^0$ term from our previous expressions for $-2\pi \I\left(\D \alpha^{\text{inst}}+\widetilde{\xi}_i^{\text{inst}}\D \xi^i -\xi^i\D \widetilde{\xi}_i^{\text{inst}}\right)$. We obtain the following:

\begin{align*}
&-\frac{R}{2\pi}\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D \rho}{2(\rho+c_{\ell})}+\frac{\D \mathcal{K}}{2} +\frac{\D \widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}\right)\\
&+\frac{R}{2\pi}\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D \rho}{2(\rho+c_{\ell})}+\frac{\D \mathcal{K}}{2} +\frac{\D \overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}\right)\\
&-R^2\sum_{\gamma}\Omega(\gamma)|\widetilde{Z}_{\gamma}|^2\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}\left(K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)+\frac{K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)}{2\pi Rn|\widetilde{Z}_{\gamma}|}\right)\left(\frac{\D \rho}{2(\rho+c_{\ell})}+\frac{\D \mathcal{K}}{2} +\frac{\D \widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}\right)\\
&-\I R\sum_{\gamma}\Omega(\gamma)\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\D \zeta_{\gamma}\\
&+R^2\sum_{\gamma}\Omega(\gamma)|\widetilde{Z}_{\gamma}|^2\left(\left(\frac{\D \rho}{2(\rho+c_{\ell})}+\frac{\D \mathcal{K}}{2} +\frac{\D \widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}\right) - \left(\frac{\D \rho}{2(\rho+c_{\ell})}+\frac{\D \mathcal{K}}{2} +\frac{\D \overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}\right)\right)\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\\
&+\I R\sum_{\gamma}\Omega(\gamma)\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\D \zeta_{\gamma}\\
&+R^2\sum_{\gamma}\Omega(\gamma)|\widetilde{Z}_{\gamma}|^2\left(\frac{\D \rho}{2(\rho+c_{\ell})}+\frac{\D \mathcal{K}}{2} +\frac{\D \overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}\right)\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}\left(K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)+\frac{K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)}{2\pi Rn|\widetilde{Z}_{\gamma}|}\right)\\
=&-\frac{R}{\pi}\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D \widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}-\frac{\D \overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}\right)\numberthis\\
\end{align*}

so comparing with (<ref>) we see that the $t^0$ terms matches $-2\I\theta_3^{P,\text{inst}}$.

Finally, we need to collect the $\D t$ term from $-2\pi \I \left(\D \alpha^{\text{inst}}+\widetilde{\xi}_i^{\text{inst}}\D \xi^i -\xi^i\D \widetilde{\xi}_i^{\text{inst}}\right)$. This one corresponds to

\begin{align*}
&-t^{-2}\frac{R}{4\pi}\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\log(1-\exp(2\pi\I \xi_{\gamma}(\zeta)))-\frac{R}{4\pi}\sum_{\gamma}\Omega(\gamma)\overline{\widetilde{Z}}_{\gamma}\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\log(1-\exp(2\pi\I \xi_{\gamma}(\zeta)))\\
&+\frac{1}{4\pi^2}\sum_{\gamma}\Omega(\gamma)\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\partial_t\left(\frac{t+\zeta}{t-\zeta}\right)\mathrm{L} (\exp(2\pi\I \xi_\gamma(\zeta)))+\frac{1}{4\pi \I }\sum_{\gamma}\Omega(\gamma)\xi_{\gamma}(t) \int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\partial_t\left(\frac{t+\zeta}{t-\zeta}\right)\log(1-\exp(2\pi\I \xi_{\gamma}(\zeta)))\\
&+\frac{1}{4\pi }\sum_{\gamma}\Omega(\gamma)\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{t+\zeta}{t-\zeta}R(-t^{-2}\widetilde{Z}_{\gamma} +\overline{\widetilde{Z}}_{\gamma})\log(1-\exp(2\pi\I \xi_{\gamma}(\zeta)))\\
=&-\frac{R}{2\pi}\sum_{\gamma}\Omega(\gamma)\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{1}{t-\zeta}(t^{-1}\widetilde{Z}_{\gamma}-\zeta\overline{\widetilde{Z}_{\gamma}})\log(1-\exp(2\pi\I \xi_{\gamma}(\zeta)))-\frac{1}{2\pi^2}\sum_{\gamma}\Omega(\gamma)\int_{l_{\gamma}}\D \zeta\frac{1}{(t-\zeta)^2}\mathrm{L} (\exp(2\pi\I \xi_\gamma(\zeta)))\\
&-\frac{1}{2\pi \I }\sum_{\gamma}\Omega(\gamma)\xi_{\gamma}(t) \int_{l_{\gamma}}\D \zeta\frac{1}{(t-\zeta)^2}\log(1-\exp(2\pi\I \xi_{\gamma}(\zeta))) \numberthis
\end{align*}

Integrating by parts the second and third term of the last equality, we obtain

\begin{align*}
&-\frac{R}{2\pi}\sum_{\gamma}\Omega(\gamma)\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{1}{t-\zeta}(t^{-1}\widetilde{Z}_{\gamma}-\zeta\overline{\widetilde{Z}_{\gamma}})\log(1-\exp(2\pi\I \xi_{\gamma}(\zeta)))\\
&+\frac{1}{2\pi}\sum_{\gamma}\Omega(\gamma)\int_{l_{\gamma}}\D \zeta\frac{1}{t-\zeta}\left(-\I \log(1-\exp(2\pi\I \xi_{\gamma}))) + \frac{\exp(2\pi\I \xi_{\gamma})}{1-\exp(2\pi\I \xi_{\gamma})}2\pi\xi_{\gamma}(\zeta)\right)(\zeta^{-2}\I R\widetilde{Z}_{\gamma}-\I R\overline{\widetilde{Z}_{\gamma}})\\
&-\sum_{\gamma}\Omega(\gamma)\xi_{\gamma}(t) \int_{l_{\gamma}}\D \zeta\frac{1}{t-\zeta}\frac{\exp(2\pi\I \xi_{\gamma})}{1-\exp(2\pi\I \xi_{\gamma})}(\zeta^{-2}\I R\widetilde{Z}_{\gamma}-\I R\overline{\widetilde{Z}_{\gamma}})\\
=&t^{-1}\frac{R}{2\pi}\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\int_{l_{\gamma}}\frac{\D \zeta}{\zeta^2}\log(1-\exp(2\pi\I \xi_{\gamma}(\zeta))) \\
&-\sum_{\gamma}\Omega(\gamma) \int_{l_{\gamma}}\D \zeta\frac{1}{t-\zeta}(\xi_{\gamma}(t)-\xi_{\gamma}(\zeta))\frac{\exp(2\pi\I \xi_{\gamma})}{1-\exp(2\pi\I \xi_{\gamma})}(\zeta^{-2}\I R\widetilde{Z}_{\gamma}-\I R\overline{\widetilde{Z}_{\gamma}})\,.\numberthis \label{appeq}
\end{align*}

Using that

\begin{equation}
\frac{1}{t-\zeta}(\xi_{\gamma}(t)-\xi_{\gamma}(\zeta))=\I R\frac{\widetilde{Z}_{\gamma}}{\zeta t}-\I R\overline{\widetilde{Z}}_{\gamma}
\end{equation}

we then obtain that the right-hand side of the equality of (<ref>) becomes

\begin{align*}
&t^{-1}\frac{R}{2\pi}\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\int_{l_{\gamma}}\frac{\D \zeta}{\zeta^2}\log(1-\exp(2\pi\I \xi_{\gamma}(\zeta))) \\
&-\sum_{\gamma}\Omega(\gamma) \int_{l_{\gamma}}\D \zeta(\I R\frac{\widetilde{Z}_{\gamma}}{\zeta t}-\I R\overline{\widetilde{Z}}_{\gamma})\frac{\exp(2\pi\I \xi_{\gamma})}{1-\exp(2\pi\I \xi_{\gamma})}(\zeta^{-2}\I R\widetilde{Z}_{\gamma}-\I R\overline{\widetilde{Z}_{\gamma}})\\
=&t^{-1}\frac{R}{2\pi}\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\int_{l_{\gamma}}\frac{\D \zeta}{\zeta^2}\log(1-\exp(2\pi\I \xi_{\gamma}(\zeta))) +t^{-1}R^2\sum_{\gamma}\Omega(\gamma) \widetilde{Z}_{\gamma}^2\int_{l_{\gamma}}\frac{\D \zeta}{\zeta^3} \frac{\exp(2\pi\I \xi_{\gamma})}{1-\exp(2\pi\I \xi_{\gamma})}\\
&-t^{-1}R^2\sum_{\gamma}\Omega(\gamma) |\widetilde{Z}_{\gamma}|^2\int_{l_{\gamma}}\frac{\D \zeta}{\zeta} \frac{\exp(2\pi\I \xi_{\gamma})}{1-\exp(2\pi\I \xi_{\gamma})}-R^2\sum_{\gamma}\Omega(\gamma) |\widetilde{Z}_{\gamma}|^2\int_{l_{\gamma}}\frac{\D \zeta}{\zeta^2} \frac{\exp(2\pi\I \xi_{\gamma})}{1-\exp(2\pi\I \xi_{\gamma})}\\
&+R^2\sum_{\gamma}\Omega(\gamma) \overline{\widetilde{Z}}_{\gamma}^2\int_{l_{\gamma}}\D \zeta \frac{\exp(2\pi\I \xi_{\gamma})}{1-\exp(2\pi\I \xi_{\gamma})} \numberthis
\end{align*}

The above integrals can be solved explicitly by using (<ref>) and (<ref>), giving

\begin{align*}
=&t^{-1}\frac{R}{\pi}\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|) \\
&+2t^{-1}R^2\sum_{\gamma}\Omega(\gamma) |\widetilde{Z}_{\gamma}|^2\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}\left(K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)+\frac{K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)}{2\pi Rn|\widetilde{Z}_{\gamma}|}\right)\\
&-2t^{-1}R^2\sum_{\gamma}\Omega(\gamma) |\widetilde{Z}_{\gamma}|^2\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\\
&-R^2\sum_{\gamma}\Omega(\gamma) |\widetilde{Z}_{\gamma}|^2\left(-2\frac{|\widetilde{Z}_{\gamma}|}{\widetilde{Z}_{\gamma}}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\right)+R^2\sum_{\gamma}\Omega(\gamma) \overline{\widetilde{Z}}_{\gamma}^2\left(-2\frac{\widetilde{Z}_{\gamma}}{|\widetilde{Z}_{\gamma}|}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\right)\\
=&t^{-1}\frac{2R}{\pi}\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\numberthis
\end{align*}

Hence, by comparing with (<ref>), we see that the $\D t$ component of $-2\pi \I\left(\D \alpha^{\text{inst}}+\widetilde{\xi}_i^{\text{inst}}\D \xi^i -\xi^i\D \widetilde{\xi}_i^{\text{inst}}\right)$ matches with $f^{\text{inst}}/t$. This completes the proof.

[1]
S. Alexandrov and S. Banerjee.
Dualities and fivebrane instantons.
Journal of High Energy Physics, 2014(11), Nov 2014.

[2]
S. Alexandrov and S. Banerjee.
Hypermultiplet metric and D-instantons.
Journal of High Energy Physics, 2015(2), 2015.

[3]
S. Alexandrov, S. Banerjee, J. Manschot, and B. Pioline.
Multiple D3-instantons and Mock modular forms I.
Communications in Mathematical Physics, 353(1):379–411, nov

[4]
S. Alexandrov, S. Banerjee, J. Manschot, and B. Pioline.
Multiple D3-instantons and Mock modular forms II.
Communications in Mathematical Physics, 359(1):297–346, mar

[5]
D. Alekseevsky, V. Cortés, and C. Devchand.
Special complex manifolds.
Journal of Geometry and Physics, 42(1-2), 2002.

[6]
D.V. Alekseevsky, V. Cortés, M. Dyckmanns, and T. Mohaupt.
Quaternionic Kähler metrics associated with special Kähler
Journal of Geometry and Physics, 92, 2015.

[7]
S. Alexandrov.
Quantum covariant c-map.
Journal of High Energy Physics, 2007(05):094–094, May 2007.

[8]
S. Alexandrov.
Twistor approach to string compactifications: A review.
Physics Reports, 522(1), 2013.

[9]
S. Alexandrov, G. Moore, A. Neitzke, and B. Pioline.
$\mathbb{R}^3$-index for four dimensional $\mathcal{N}=2$ field
Physical Review Letters, 114(12), 2015.

[10]
S. Alexandrov, J. Manschot, D. Persson, and B. Pioline.
Quantum hypermultiplet moduli spaces in ${N}=2$ string vacua: a
String-Math 2012 Proceedings of Symposia in Pure Mathematics,

[11]
S. Alexandrov and B. Pioline.
S-duality in twistor space.
Journal of High Energy Physics, 2012(8), Aug 2012.

[12]
S. Alexandrov, D. Persson, and B. Pioline.
Wall-crossing, Rogers dilogarithm and the QK-HK correspondence.
Journal of High Energy Physics, 2011(12), 2011.

[13]
S. Alexandrov, B. Pioline, F. Saueressig, and S. Vandoren.
D-instantons and twistors.
Journal of High Energy Physics, 2009(03):044–044, Mar 2009.

[14]
S. Alexandrov, B. Pioline, F. Saueressig, and S. Vandoren.
Linear perturbations of quaternionic metrics.
Communications in Mathematical Physics, 296(2):353–403, Feb

[15]
S. Alexandrov and F. Saueressig.
Quantum mirror symmetry and twistors.
Journal of High Energy Physics, 2009(09):108–108, sep 2009.

[16]
R. Böhm, H. Günther, C. Herrmann, and J. Louis.
Compactification of type IIB string theory on Calabi–Yau
Nuclear Physics B, 569(1-3):229–246, Mar 2000.

[17]
T. Bridgeland.
Riemann-Hilbert problems from Donaldson-Thomas theory.
Inventiones mathematicae, 216, 2019.

[18]
V. Cortés, M. Dyckmanns, M. Jüngling, and D. Lindemann.
A class of cubic hypersurfaces and quaternionic Kähler manifolds
of co-homogeneity one.
Asian Journal of Mathematics, 25:1–30, 2020.

[19]
V. Cortés, P. Dieterich, and T. Mohaupt.
ASK/PSK-correspondence and the r-map.
Lett. Math. Phys., 108(5):1279–1306, 2018.

[20]
V. Cortés, M. Dyckmanns, and S. Suhr.
Completeness of projective Kähler and quaternionic Kähler
Springer INdAM Series Special Metrics and Group Actions in
Geometry, 23, 2017.

[21]
S. Cecotti.
Special geometry and the swampland.
Journal of High Energy Physics, 2020(9), sep 2020.

[22]
S. Cecotti, S. Ferrara, and L. Girardello.
Geometry of type II superstrings and the moduli of superconformal
field theories.
Journal of Modern Physics A, 4(10), 1989.

[23]
V. Cortés, X. Han, and T. Mohaupt.
Completeness in supergravity constructions.
Communications in Mathematical Physics, 311(1):191–213, Feb

[24]
V. Cortés.
On hyper Kähler manifolds associated to Lagrangian Kähler
submanifolds of $T^*\mathbb{C}^n$.
Trans. Amer. Math. Soc., 350(8), 1998.

[25]
V. Cortés and I. Tulli.
Quaternionic Kähler metrics associated to special Kähler
manifolds with mutually local variations of BPS structures.
Annales Henri Poincaré, 23:2025–2067, Jan 2022.

[26]
V. Cortés and I. Tulli.
S-duality and the universal isometries of q-map spaces.
Comm. Math. Phys., 394:833–885, June 2022.

[27]
B. de Wit and A. Van Proeyen.
Special geometry, cubic polynomials and homogeneous quaternionic
Communications in Mathematical Physics, 149(2):307 – 333,

[28]
B. de Wit and A. Van Proeyen.
Isometries of special manifolds.
Proceedings of the Meeting on Quaternionic Structures in
Mathematics and Physics, Trieste, September 1994, ILAS/FM-6/1996:109–134,

[29]
B. de Wit, F. Vanderseypen, and A. Van Proeyen.
Symmetry structure of special geometries.
Nuclear Physics B, 400(1–3):463–521, Jul 1993.

[30]
B. Forbes and M. Jinzenji.
Extending the Picard-Fuchs system of local mirror symmetry.
Journal of Mathematical Physics, 46(8):082302, aug 2005.

[31]
D. Freed.
Special Kähler manifolds.
Communications in Mathematical Physics, 203(1), 1999.

[32]
S. Ferrara and S. Sabharwal.
Quaternionic manifolds for type II superstring vacua of
Calabi-Yau spaces.
Nuclear Physics B, 332(2), 1990.

[33]
D. Gaiotto, G. Moore, and A. Neitzke.
Four-dimensional wall-crossing via three-dimensional field theory.
Communications in Mathematical Physics, 299(1), 2010.

[34]
D. Joyce and Y. Song.
A Theory of generalized Donaldson-Thomas invariants.
Memoirs of the A.M.S, 10 2008.

[35]
C. LeBrun.
Quaternionic-Kähler manifolds and conformal geometry.
Mathematische Annalen, 284(3):353–376, 1989.

[36]
A. Neitzke, B. Pioline, and S. Vandoren.
Twistors and black holes.
Journal of High Energy Physics, 2007(04):038–038, Apr 2007.

[37]
H. Ooguri and C. Vafa.
On the geometry of the string landscape and the swampland.
Nuclear Physics B, 766(1-3):21–33, mar 2007.

[38]
D. Robles-Llana, F. Saueressig, and S. Vandoren.
String loop corrected hypermultiplet moduli spaces.
J. High Energy Phys, 81(3), 2006.

[39]
S. Salamon.
Quaternionic Kähler manifolds.
Inventiones mathematicae, 67:143–172, 1982.