Zoo of flows in a 3d gauged supergravity with periodic potential

TL;DR

This work investigates holographic RG flows in a truncated $D=3$, $\mathcal{N}=(2,0)$ gauged supergravity with a periodic scalar potential, focusing on interpolations from AdS/dS to Minkowski and extending to finite temperature. The authors recast the equations of motion as autonomous dynamical systems on multiple phase spaces, deriving exact domain-wall and black-string solutions that correspond to irrelevant-operator VEV deformations of the dual 2d CFTs. They show that at finite temperature most flows are singular, with regular geometries restricted to BTZ and Schwarzschild–de Sitter black holes, and they provide analytic near-horizon descriptions. The results illuminate exotic holographic RG flows in low dimensions and point toward uplift to string/M-theory contexts and higher-dimensional generalizations as promising future work.

Abstract

In this paper we construct solutions with AdS/dS asymptotics for $D=3$ truncated gauged supergravity with a periodic scalar potential. In a holographic perspective assuming Dirichlet boundary conditions, the solutions can be interpreted as deformations of 2d dual CFTs triggered by non-zero vacuum expectation values of irrelevant operators. In addition to the domain wall type solutions, we incorporated in the analysis a black string solution, which can be also interpreted as a deformation by VEV of an irrelevant operator. Generalizing the flows to finite temperature we find that the corresponding geometries are singular but have horizons. For certain flows we provide an analytical description near the horizon region.

Figures (8)

• Figure 1: The behavior of the potential \ref{['cospot']} for $a^2=1$ and $m=1$ on the interval $[-\pi,\pi]$.
• Figure 2: Plot of the superpotential \ref{['cossuppot']} with $m=1$. There are three $AdS$ and two Minkowski $M$ critical points on the interval $\phi\in [-\pi,\pi]$.
• Figure 3: Phase portrait of the supersymmetric system \ref{['first']}, with $m=1$, $a^2=1$. Red and blue trajectories correspond to the same $n=0$, but to opposite signs for the dilaton field, together with different constants $\mathrm{c_A}$ for the scale factor of the exact solution \ref{['Degsol']}.
• Figure 4: Phase portrait and numerical trajectories on the period $\phi\in(-\pi,\pi)$. We set $a^2=1$. Orange and green curves from the $AdS$ point are two branches of supersymmetric explicit solution \ref{['Degsol']}. Red and purple curves are the numerical solutions of the system \ref{['cossys']} with initial conditions $(\phi,X) = (-0.955,\pm 0.025)$.
• Figure 5: Global phase portrait for the system \ref{['sysinD']} with \ref{['syscompdiskmain']} in the disk $\mathbb{D}^2$.