- |\phi_3|^2\,\big)\;, 
\end{split}$$ where $\phi_a$ are the three complex adjoint scalars of $\mathcal{N}=4$ SYM.[^5] The last factor in is parametrized by the complex scalar $\chi e^{i\theta}$ in the $\bf 10\oplus\overline{10}$ of $SO(6)$, plus the dilaton and axion dual to the complexified gauge coupling of $\mathcal{N}=4$ SYM. For the solutions we are interested in, one can consistently turn off the dilaton and axion and set the phase $\theta$ to be constant. This leaves only one real scalar, $\chi$, in the third factor in , which is dual to the fermion bilinear $$\label{Ochi}
\mathcal{O}_{\chi} = {\rm Tr} (\chi_1\chi_1 + {\rm h.c.})\;,$$ where, as in Section \[subsec:ANSYM\], $\chi_a$ are the adjoint Weyl fermions in the three chiral super fields.

The bosonic part of the action in this sector is derived in Appendix \[appendixA\]. For the trivial dilaton/axion, it reads[^6] $$\label{CCaction}
\begin{split}
e^{-1}\cals L&  \eql  -{1\over 4} R -{1\over 4}\Big[e^{4(\alpha-\beta)}\,F^{(1)}_{\mu\nu}F^{(1)}{}^{\mu\nu}+e^{4(\alpha+\beta)} F^{(2)}_{\mu\nu}F^{(2)}{}^{\mu\nu}+e^{-8\alpha}F^{(3)}_{\mu\nu}F^{(3)}{}^{\mu\nu}\Big]
\\[5pt] & +3(\partial_\mu\alpha)^2+(\partial_\mu\beta)^2+{1\over 2}(\partial_\mu\chi)^2+{1\over 8}\sinh^2(2\chi)\Big[\partial_\mu\theta+{g}\,(A^{(1)}+A^{(2)}-A^{(3)})\Big]^2 -g^2\,\cals P\,,
\end{split}$$ with the scalar potential $$\label{CCpotential}
\begin{split}
\cals P\eql {1\over 8}e^{-4(\alpha+\beta)}\cosh^2\chi\Big[
\big(1+e^{8\beta}+e^{4(3\alpha+\beta)}\big)\cosh^2\chi-\big(1+e^{4\beta}+e^{2(3\alpha+\beta)}\big)^2
\Big]\,.
\end{split}$$ As in similar supersymmetric truncations (see, e.g. [@Freedman:1999gp; @Bobev:2010de]), the potential can be rewritten as $$\mathcal{P} =\frac{1}{48} \left(\partial_{\alpha}\mathcal{W}\right)^2 + \frac{1}{16} \left(\partial_{\beta}\mathcal{W}\right)^2 + \frac{1}{8} \left(\partial_{\chi}\mathcal{W}\right)^2- \frac{1}{3}\mathcal{W}^2\;,$$ where $$\label{suppot5d}
\cals  W\eql \frac{1}{4} e^{-2 (\alpha 