Holographic RG flows and symplectic deformations of N=4 gauged supergravity

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We study four-dimensional $N=4$ gauged supergravity with $SO(4)\times SO(4)\sim SO(3)\times SO(3)\times SO(3)\times SO(3)$ gauge group in the presence of symplectic deformations. There are in general four electric-magnetic phases corresponding to each $SO(3)$ factor, but two phases of the $SO(3)$ factors embedded in the $SO(6)$ R-symmetry are fixed. One phase can be set to zero by $SL(2,\mathbb{R})$ transformations. The second one gives equivalent theories for any non-vanishing values and can be set to $\frac{\pi}{2}$ resulting in gauged supergravities that admit $N=4$ supersymmetric $AdS_4$ vacua. The remaining two phases are truely deformation parameters leading to different $SO(4)\times SO(4)$ gauged supergravities. As in the $\omega$-deformed $SO(8)$ maximal gauged supergravity, the cosmological constant and scalar masses of the $AdS_4$ vacuum at the origin of the scalar manifold with $SO(4)\times SO(4)$ symmetry do not depend on the electric-magnetic phases. We find $N=1$ holographic RG flow solutions between $N=4$ critical points with $SO(4)\times SO(4)$ and $SO(3)_{\textrm{diag}}\times SO(3)\times SO(3)$ or $SO(3)\times SO(3)_{\textrm{diag}}\times SO(3)$ symmetries. We also give $N=2$ and $N=1$ RG flows from these critical points to various non-conformal phases. However, contrary to the $\omega$-deformed $SO(8)$ gauged supergravity, there exist non-trivial supersymmetric $AdS_4$ critical points only for particular values of the deformation parameters within the scalar sectors under consideration.