Three-Dimensional SCFTs, Supersymmetric Domain Wall and Renormalization Group Flow

TL;DR

This work analyzes a $SU(3)\times U(1)$-invariant stationary point of gauged $N=8$ supergravity and shows that deforming the ${\bf S}^7$ compactification induces a holographic RG flow in the dual 3D SCFT from a maximal $SO(8)$ UV fixed point to an $\mathcal{N}=2$ $SU(3)\times U(1)$ IR fixed point. By constructing the explicit $28$-bein variables $u,v$ in the $56$-dimensional representation of $E_7$, the authors derive the $A_1,A_2$ tensors and identify an eigenvalue of $A_1$ with the superpotential $W$, enabling a 4D BPS domain-wall description of the flow. They compute the mass spectra of the scalar fluctuations around both fixed points to determine the conformal dimensions of the perturbing operator: the deformation is relevant at the UV fixed point ($\Delta=1$ or $2$) and irrelevant at the IR fixed point ($\Delta \approx 4.56$). A numerical steepest-descent integration of the BPS equations yields a smooth domain-wall interpolating between the two AdS$_4$ vacua, illustrating a concrete holographic realization of a 3D SCFT RG flow with explicit supergravity data. The results provide a detailed map between bulk deformations and boundary operator dimensions, with explicit machinery for exploring AdS/CFT in lower dimensions.

Abstract

By analyzing SU(3)xU(1) invariant stationary point, studied earlier by Nicolai and Warner, of gauged N=8 supergravity, we find that the deformation of S^7 gives rise to nontrivial renormalization group flow in a three-dimensional boundary super conformal field theory from N=8, SO(8) invariant UV fixed point to N=2, SU(3)xU(1) invariant IR fixed point. By explicitly constructing 28-beins u, v fields, that are an element of fundamental 56-dimensional representation of E_7, in terms of scalar and pseudo-scalar fields of gauged N=8 supergravity, we get A_1, A_2 tensors. Then we identify one of the eigenvalues of A_1 tensor with ``superpotential'' of de Wit-Nicolai scalar potential and discuss four-dimensional supergravity description of renormalization group flow, i.e. the BPS domain wall solutions which are equivalent to vanishing of variation of spin 1/2, 3/2 fields in the supersymmetry preserving bosonic background of gauged N=8 supergravity. A numerical analysis of the steepest descent equations interpolating two critical points is given.

Figures (3)

• Figure 1: Scalar potential $V(\lambda, \lambda')$. The left axis corresponds to $\lambda$ and right one does $\lambda'$. The extremum value $V=-9\sqrt{3}/2=-7.79$ for $SU(3) \times U(1)$ occurs around $\lambda=0.78$ and $\lambda'=0.93$ while the local maximum value $V=-6$ for $SO(8)$ appears around $\lambda=0$ and $\lambda'=0$. We take $g^2$ as 1 for simplicity.
• Figure 2: The contour map of $V$ (on the left) and $W$ (on the right), with $\lambda'$ on the vertical axis and $\lambda$ on the horizontal axis. $V$ has vanishing first derivatives in all directions orthogonal to the plane. At $(\lambda, \lambda')=(0, 0)$, it is the maximally supersymmetric and locally maximum of $V$ while minimum of $W$. At $(\lambda, \lambda')=(0.78, 0.93)$, it is ${\cal N}=2$ supersymmetric and other extremum of both $V$ and $W$.
• Figure 3: The plots of $e^{\frac{\lambda}{2\sqrt{2}}}$ (on the left), $e^{\frac{\lambda'}{2\sqrt{2}}}$ (on the middle) and $\partial_r A$ (on the right), with $\tanh r$ on the horizontal axis. They arrive at 1, 1, 2.40 respectively when $r \rightarrow \infty$ or $\tanh r|_{r \rightarrow \infty} =1$. The value of $\lambda =0=\lambda'$ corresponds to the expectation values of fields of UV fixed point. The asymptotic value of last one is consistent with the vlaue of $\partial_r A = \sqrt{2} g W$ at UV fixed point where $W =1$. We took $g=1.7$ for simplicity.