$M\in\mathrm{SL}(2,\fieldR)$ as $u_+
\rightarrow M u_+$, $u_-\rightarrow (M^{-1})^tu_-$. We have therefore identified a potential in terms of real coordinates which determines the metric components. Such spaces are called para-Kähler[^6]. The metric for the Minkowskian universal hypermultiplet moduli space is of the same form as in , but with $u^i_\pm$ treated as complex coordinates, where $u^i_-=-
\bar{u}^i_+$ under complex conjugation.

Determination of O(2) matrices {#appO2}
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We need to solve the differential equations $$\begin{aligned}
  0 & = {\partial}_\phi c - {\mathrm{e}}^{\phi/2} {\partial}_\chi s \label{O2_1} \\[2pt]
  0 & = {\partial}_\phi s + {\mathrm{e}}^{\phi/2} {\partial}_\chi c - {\tfrac{1}{2}}(2{\epsilon}- 1) s\ ,
    \label{O2_2}
 \end{aligned}$$ where $c$ and $s$ are subject to the constraint $c^2+s^2=1$. We first multiply by $s$ and by $c$, respectively, and use $-s{\partial}s=c{\partial}c$ to write the equations as $$\begin{aligned}
  0 & = s\, {\partial}_\phi c + {\mathrm{e}}^{\phi/2} c\, {\partial}_\chi c \label{O2_3} \\[2pt]
  0 & = c\, {\partial}_\phi s + {\mathrm{e}}^{\phi/2} c\, {\partial}_\chi c - {\tfrac{1}{2}}(2{\epsilon}- 1)
    c s\ . \label{O2_4}
 \end{aligned}$$ Multiplying the difference of these equations by $c$ gives $$\label{O2_5}
  0 = c \big[ c\, {\overset{\leftrightarrow}{{\partial}_\phi}} s - {\tfrac{1}{2}}(2{\epsilon}- 1) c s \big] =
  {\partial}_\phi s - {\tfrac{1}{2}}(2{\epsilon}- 1)\, (1 - s^2) s\ ,$$ which involves only $s$ and can 