In this paper, we consider finite-temperature holographic renormalization group (RG) flows in $D=3$ $\mathcal{N}=(2,0)$ gauged truncated supergravity coupled to a sigma model with a hyperbolic target space. In the context of the holographic duality, fixed points (CFTs) at finite temperature are described by anti–de Sitter (AdS) black holes. We come from the gravity equations of motion to a 3D autonomous dynamical system, which critical points can be related to fixed points of dual field theories. Near-horizon black hole solutions correspond to infinite points of this system. We use Poincaré transformations to project the system on ${\mathbf{R}}^3$ into the 3D unit cylinder such that the infinite points are mapped onto the boundary of the cylinder. We explore numerically the space of solutions. We show that the exact RG flow at zero temperature is the separatrix for asymptotically AdS black hole solutions if the potential has one extremum, while for the potential with three extrema the separatrices are RG flows between AdS fixed points. We find near-horizon analytical solutions for asymptotically AdS black holes using the dynamical equations. We also present a method for constructing full analytical solutions.

中文翻译：

---

全息 RG 在有限温度下的 3D 测量超重力中流动


在本文中，我们考虑有限温度全息重整化群（RG）流 $D=3$ $\mathcal{N}=(2,0)$ 测量截断超重力耦合到具有双曲目标空间的 sigma 模型。在全息对偶性的背景下，有限温度下的不动点（CFT）由反德西特（AdS）黑洞描述。我们从运动的重力方程转向 3D 自主动力系统，其临界点可以与对偶场论的不动点相关。近地平线黑洞解对应于该系统的无限点。我们使用庞加莱变换将系统投影到 ${\mathbf{R}}^3$ 到 3D 单位圆柱体中，以便将无限点映射到圆柱体的边界上。我们以数字方式探索解决方案的空间。我们证明，如果势具有一个极值，则零温度下的精确 RG 流是渐近 AdS 黑洞解的分界线，而对于具有三个极值的势，分界线是 AdS 固定点之间的 RG 流。我们使用动力学方程找到了渐近 AdS 黑洞的近地平线解析解。我们还提出了一种构建完整分析解决方案的方法。