Holographic RG flows in a 3D gauged supergravity at finite temperature

In this paper, we consider finite-temperature holographic renormalization group (RG) flows in $D=3$ $N=(2,0)$ 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 $R^3$ 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.