Perturbative Hairy Black Branes in the $G_{2}$-Invariant Sector of Dyonic ISO(7) Gauged Supergravity

Abstract: We construct perturbative black-brane solutions carrying neutral scalar hair in the $G_{2}$-invariant truncation of four-dimensional dyonic $\mathcal{N}=8$ ISO(7) gauged supergravity, an exact consistent truncation of massive type~IIA supergravity on $S<sup>{6}$. Expanding around the $G_{2}$-symmetric AdS$<em>{4}$-Schwarzschild black brane in powers of a dimensionless scalar-charge parameter $\eps$, we show that the perturbation hierarchy reduces, order by order, to a nested sequence of inhomogeneous Legendre equations (demonstrated through second order). The first-order scalar satisfies a Pöschl--Teller equation with exact parameter $ν(ν+1)=2/3$, giving $ν=-\frac{1}{2}+\frac{\sqrt{33}}{6}$, and the first-order metric is fixed algebraically by the same function, $A<sup>{(1)}=-\tfrac{\sqrt{7}}{2}φ<sup>{(1)}$, so the entire $\mathcal{O}(\eps)$ sector is one Legendre function. An identity $ν+1=Δ</em>{+}/3$, with $Δ<em>{+}=(3+\sqrt{33})/2\approx 4.372$ the conformal dimension of the dual scalar operator, relates the Pöschl-Teller parameter to the AdS/CFT dictionary via the nonlinear Fefferman-Graham map $z \propto z</em>{\mathrm{FG}}<sup>{3}$. The $\mathcal{O}(\eps<sup>{2})$ back-reaction yields explicit corrections to the renormalized free energy and Bekenstein-Hawking entropy density at fixed temperature, consistent with the first law; equivalently, at fixed entropy or energy density it \emph{lowers} the Hawking temperature by $δT/T=\mathcal{O}(\eps<sup>{2})$. For the leading transport coefficients the shear viscosity saturates $η/s=1/(4π)$, while the Eling-Oz horizon formula gives a nonzero bulk viscosity $ζ/η=\bigl(\tfrac{Δ<em>{+}-3}{2}\bigr)<sup>{2}\eps<sup>{2}=\tfrac{3(7-\sqrt{33})}{8}\eps<sup>{2}\approx0.471\,\eps<sup>{2}$. With $c</em>{s}^{{2}=\tfrac12-\mathcal} O(\eps<sup>{2})&lt;\tfrac12$, this indicates a softening of the equation of state.