Evidence for the Holographic dual of $\mathcal{N}=3$ Solution in Massive Type IIA

We calculate the Kaluza-Klein spectrum of spin-2 fluctuations around the $\mathcal{N}=3$ warped ${\mathrm{A}\mathrm{d}\mathrm{S}}_4\times M_6$ solution in massive IIA supergravity. This solution was conjectured to be dual to the $D=3$ ${\cal N}=3$superconformal${\rm SU}(N)$Chern-Simonsmattertheorywithlevel$k$and2adjointchiralmultiplets\mathrm{.}The${\rm SO}(3)_R\times{\rm SO}(3)_D$isometryofthe${\cal N}=3$solutionisidentifiedwiththe${\rm SU}(2)F\times {\rm SU}(2){\cal R}$globalsymmetryofthedual${\cal N}=3$SCFT\mathrm{.}Weshowthatthe${\rm SO}(3)_R\times{\rm SO}(3)_D$ quantum numbers and the $\mathrm{A}\mathrm{d}\mathrm{S}$ energies carried by the BPS spin-2 modes match precisely with those of the spin-2 gauge invariant operators in the short multiplets of operators in the $\mathcal{N}=3$ SCFT. We also compute the Euclidean action of the $\mathcal{N}=3$ solution and the free energy of the $\mathcal{N}=3$ SCFT on $S^3$, in the limit $N\gg k$. Remarkably, the results show a complete agreement.