s + {\mathrm{e}}^{\phi/2} {\partial}_\chi c - {\tfrac{1}{2}}(2{\epsilon}- 1) s\ .
 \end{aligned}$$ We derive the general solution in appendix \[appO2\]. The result is that there are *three* distinct BPS conditions corresponding to the O(2) matrices $$O_{1,2} = \pm \begin{pmatrix} 1 & 0 \\[2pt] 0 & {\epsilon}\end{pmatrix}\
  ,\quad O_3 = \pm {\frac{\raisebox{-2pt}{$1$}}{|\tau'|}} \begin{pmatrix} \operatorname{Re}\tau' & -\operatorname{Im}\tau'\, \\[4pt] \operatorname{Im}\tau' & \operatorname{Re}\tau' \end{pmatrix}\ ,$$ invariant under both and . Here $\tau'=\tau-\chi_0$ with $\tau$ as in , $\chi_0$ is a real integration constant, and the plus and minus signs refer to the instanton and anti-instanton, respectively.

For these three O(2) matrices the 1-form $N^t OE$ is exact, $$\label{dY}
  N^t O E = \pm\, {\mathrm{d}}Y\ ,$$ where modulo an additive constant $$Y_{1,2} = \begin{pmatrix} {\epsilon}\chi \\[2pt] {\mathrm{e}}^\phi - {\tfrac{1}{2}}{\epsilon}\chi^2
  \end{pmatrix}\ ,\quad Y_3 = {\tfrac{1}{2}}\sqrt{4 {\mathrm{e}}^{\phi} + (\chi -
  \chi_0)^2\,} \begin{pmatrix} 2 \\ - \chi - \chi_0 \end{pmatrix}\ .$$ It follows that the action for BPS configurations is given by a topological boundary term $$\label{top-BT}
  S^E\,|_\mathrm{BPS} = (-)^D \int_{\mathcal{M}}H^t {\wedge}N^t O E = \mp \int_{{\partial}{\mathcal{M}}}
  Y^t H\ .$$ The instanton action is therefore determined by the charges $Q^{(a)}$ and the values of the fields $\chi$ and ${\mathrm{e}}^{\phi}$ at the boundaries.

It is easy to find the corresponding 