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Supersymmetric Domain Wall and RG Flow from 4-Dimensional Gauged N=8 Supergravity

Changhyun Ahn, Kyungsung Woo

TL;DR

This work analyzes RG-flow physics in the AdS$_4$/CFT$_3$ context by examining invariant sectors of gauged ${\cal N}=8$ supergravity in four dimensions. It identifies a superpotential $W$ from specific eigenvalues of the ${A_1}^{IJ}$ tensor across SU(3), SO(5), and SO(3)×SO(3) sectors and derives BPS domain-wall equations that describe holographic RG flows between UV and IR AdS$_4$ vacua. The paper provides explicit scalar potentials and first-order flow equations, including a rare analytic non-BPS domain-wall in the ${\rm SO}(3)\times{\rm SO}(3)$ sector, and connects these flows to known eleven-dimensional embeddings via the ${\bf S}^7$ deformation. Collectively, these results map how relevant operators drive holographic RG flows between maximally symmetric and less supersymmetric fixed points in the dual 3D theory, clarifying the role of sector-specific superpotentials in the dynamics of gauged ${\cal N}=8$ supergravity.

Abstract

By studying various, known extrema of 1) SU(3) sectors, 2) SO(5) sectors and 3) SO(3)xSO(3) sectors of gauged N =8 supergravity in four-dimensions, one finds that the deformation of seven sphere §^7 gives rise to non-trivial renormalization group(RG) flow in three-dimensional boundary conformal field theory from UV fixed point to IR fixed point. For SU(3) sectors, this leads to four-parameter subspace of the supergravity scalar-gravity action and we identify one of the eigenvalues of A_1 tensor of the theory with a superpotential of scalar potential that governs RG flows on this subspace. We analyze some of the structure of the superpotential and discuss first-order BPS domain-wall solutions, using some algebraic relations between superpotential and derivatives of it with respect to fields, that determine a (super)symmetric kink solution in four-dimensional N =8 supergravity, which generalizes all the previous considerations. The BPS domain-wall solutions are equivalent to vanishing of variation of spin 1/2, 3/2 fields in the supersymmetry preserving bosonic background of gauged N=8 supergravity. For SO(5) sectors, there exist only nontrivial nonsupersymmetric critical points that are unstable and included in SU(3) sectors. For SO(3)xSO(3) sectors, we construct the scalar potential(never been written) explicitly and study explicit construction of first-order domain-wall solutions.

Supersymmetric Domain Wall and RG Flow from 4-Dimensional Gauged N=8 Supergravity

TL;DR

This work analyzes RG-flow physics in the AdS/CFT context by examining invariant sectors of gauged supergravity in four dimensions. It identifies a superpotential from specific eigenvalues of the tensor across SU(3), SO(5), and SO(3)×SO(3) sectors and derives BPS domain-wall equations that describe holographic RG flows between UV and IR AdS vacua. The paper provides explicit scalar potentials and first-order flow equations, including a rare analytic non-BPS domain-wall in the sector, and connects these flows to known eleven-dimensional embeddings via the deformation. Collectively, these results map how relevant operators drive holographic RG flows between maximally symmetric and less supersymmetric fixed points in the dual 3D theory, clarifying the role of sector-specific superpotentials in the dynamics of gauged supergravity.

Abstract

By studying various, known extrema of 1) SU(3) sectors, 2) SO(5) sectors and 3) SO(3)xSO(3) sectors of gauged N =8 supergravity in four-dimensions, one finds that the deformation of seven sphere §^7 gives rise to non-trivial renormalization group(RG) flow in three-dimensional boundary conformal field theory from UV fixed point to IR fixed point. For SU(3) sectors, this leads to four-parameter subspace of the supergravity scalar-gravity action and we identify one of the eigenvalues of A_1 tensor of the theory with a superpotential of scalar potential that governs RG flows on this subspace. We analyze some of the structure of the superpotential and discuss first-order BPS domain-wall solutions, using some algebraic relations between superpotential and derivatives of it with respect to fields, that determine a (super)symmetric kink solution in four-dimensional N =8 supergravity, which generalizes all the previous considerations. The BPS domain-wall solutions are equivalent to vanishing of variation of spin 1/2, 3/2 fields in the supersymmetry preserving bosonic background of gauged N=8 supergravity. For SO(5) sectors, there exist only nontrivial nonsupersymmetric critical points that are unstable and included in SU(3) sectors. For SO(3)xSO(3) sectors, we construct the scalar potential(never been written) explicitly and study explicit construction of first-order domain-wall solutions.

Paper Structure

This paper contains 12 sections, 102 equations, 1 figure.

Table of Contents

  1. Introduction
  2. de Wit-Nicolai Potential
  3. $SU(3)$ Sectors of Gauged ${\cal N}=8$ Supergravity
  4. $SO(5)$ Sectors of Gauged ${\cal N}=8$ Supergravity
  5. $SO(3) \times SO(3)$ Sectors of Gauged ${\cal N}=8$ Supergravity
  6. Supersymmetric Domain Wall and RG Flow
  7. $SU(3)$ Sectors
  8. $SO(5)$ Sector
  9. $SO(3) \times SO(3)$ Sectors
  10. Appendix A: $SU(3)$ Invariant Sectors
  11. Appendix B: $SO(5)$ Invariant Sectors
  12. Appendix C: $SO(3) \times SO(3)$ Invariant Sector

Figures (1)

  • Figure 1: The plots of $V$ (on the left) and $W$ (on the right), with $\lambda^1$ on the horizontal axis. Scalar potential $V$, at $\lambda^1=0$, is the maximally supersymmetric and locally maximum while superpotential $W$ at that point is locally minimum. The cosmological constant is $-6$. We took $g^2$ as 1 for simplicity. At $\lambda^1=2 \sinh^{-1} 2=2.89$, $V$ has locally minimum and is nonsupersymmetric and the cosmological constant is $-14$. See also distler.