*\! H +
  O E \big)^t {\wedge}\big( N\! *\! H + O E \big) + (-)^D H^t {\wedge}N^t O E\ .$$ Here we have defined $$H = \begin{pmatrix} H^1 \\[2pt] H^2 \end{pmatrix}\ ,\quad E =
  \begin{pmatrix} {\mathrm{d}}\phi \\[2pt] {\mathrm{e}}^{-\phi/2}\, {\mathrm{d}}\chi
 \end{pmatrix}\ ,\quad N = {\mathrm{e}}^{\phi/2} \begin{pmatrix} 0 & {\mathrm{e}}^{\phi/2}\,
  \\[2pt] 1 & -\chi \end{pmatrix}\ ,$$ such that $N^t N=M$, and $O$ is some orthogonal (scalar) field-dependent matrix, whose appearance is due to the fact that $N$ and the zweibein $E$ are determined only modulo local O(2) transformations.

Clearly, the action is bounded from below by $$\label{bound}
  S^E \geq \int_{{\mathcal{M}}} \big( {\mathrm{d}}^Dx\, \sqrt{g\,} R + (-)^D H^t {\wedge}N^t
  O E \big)\ ,$$ where the second term is topological, as it is independent of the spacetime metric. The bound is saturated by field configurations satisfying the BPS condition $$\label{BPS}
  * H = - N^{-1} O E\ .$$ A similar Bogomol’nyi equation was derived for an $N=1$, $D=4$ tensor multiplet (containing one tensor and one scalar) in [@R]. Notice that, if the matrix $O$ is invariant, this equation transforms covariantly under and .

Equation is a proper BPS condition only if it implies the equations of motion, and this will fix the O(2) 