Symmetric Potentials of Gauged Supergravities in Diverse Dimensions and Coulomb Branch of Gauge Theories

TL;DR

The paper analyzes symmetric scalar potentials within gauged supergravities across dimensions $D=4,5,7$, showing that scalar sectors on the $SL(N,\mathbb{R})/SO(N)$ coset are governed by a universal potential and yield AdS domain-wall geometries. These domain walls lift to higher-dimensional distributions of D3-, M2-, M5-branes and the D4-D8 system, corresponding to Coulomb-branch states in the dual CFTs, with explicit harmonic-function descriptions and charge distributions. A consistent KK reduction framework is established, linking lower-dimensional potentials to higher-dimensional supergravity via a unified reduction Ansatz that preserves the scalar dynamics and enables the interpretation of domain walls as distributed branes. The spectrum of linear fluctuations around these backgrounds is studied in detail, revealing a universal pattern: Euclidean parameters produce discrete spectra for $n_e=1,2,3$ and continuous spectra (with/without a gap) for higher $n_e$, while Lorentzian cases (notably D4-D8) exhibit complementary behavior and relate to the Euclidean cases through parameter mappings. Overall, the work connects symmetric scalar potentials, domain-wall geometries, and brane distributions across dimensions, highlighting the Coulomb-branch physics and the stability of the linear spectra.

Abstract

A class of conformally flat and asymptotically anti-de Sitter geometries involving profiles of scalar fields is studied from the point of view of gauged supergravity. The scalars involved in the solutions parameterise the SL(N,R)/SO(N) submanifold of the full scalar coset of the gauged supergravity, and are described by a symmetric potential with a universal form. These geometries descend via consistent truncation from distributions of D3-branes, M2-branes, or M5-branes in ten or eleven dimensions. We exhibit analogous solutions asymptotic to AdS_6 which descend from the D4-D8-brane system. We obtain the related six-dimensional theory by consistent reduction from massive type IIA supergravity. All our geometries correspond to states in the Coulomb branch of the dual conformal field theories. We analyze linear fluctuations of minimally coupled scalars and find both discrete and continuous spectra, but always bounded below.

Figures (7)

• Figure 1: The Schrödinger potentials $V(z)$ with $n_e=1$ Euclidean parameters are given for M2-branes, D3-branes, D4-D8-branes and M5-branes, with successively increasing values of the potential. The additional solid line represents the D4-D8-brane with $n=4$ equal Lorentz parameters. The additive ambiguity for $z$ is fixed in such a way that $z=0$ corresponds to the horizon boundary, and $z$ is rescaled so that $z=1$ is the boundary of AdS in all cases.
• Figure 2: The Schrödinger potentials $V(z)$ with $n_e=2$ Euclidean parameters are given for M2-branes, D3-branes, D4-D8-branes and M5-branes, with successively increasing values of the potential. The additional solid line represents the D4-D8-brane with $n=3$ equal Lorentz parameters. The additive ambiguity for the $z$ is fixed in such a way that $z=0$ corresponds to the horizon boundary, and $z$ is rescaled so that $z=1$ is the boundary of AdS in all cases.
• Figure 3: The Schrödinger potentials $V(z)$ with $n_e=3$ Euclidean parameters are given for M2-branes, D3-branes, D4-D8-branes and M5-branes, with successively increasing values of the potential. The additional solid line represents the D4-D8-brane with $n=2$ equal Lorentz parameters. The additive ambiguity for $z$ is fixed in such a way that $z=0$ corresponds to the horizon boundary, and $z$ is rescaled so that $z=1$ is the boundary of AdS in all cases.
• Figure 4: The Schrödinger potentials $V(z)$ with $n_e=4$ Euclidean parameters are given for M2-branes, D3-branes, D4-D8-branes and M5-branes, with successively increasing values of the potential. The additional solid line represents the D4-D8-brane with $n=1$ Lorentz parameter. $z=0$ corresponds to the AdS boundary. $V\to 1$ for the $n_e=4$ cases and $V\to 0$ for $n=1$.
• Figure 5: The Schrödinger potentials $V(z)$ with $n_e=5$ Euclidean parameters are given for M2-branes and D3-branes, with successively increasing values of the potential. The AdS boundary is at $z=0$.