AdS black holes in two-dimensional dilaton gravity and holography

In this paper, we present two novel analytic anti–de Sitter (AdS) black hole solutions in a two-dimensional dilaton gravity theory with two scalar fields nonminimally coupled to gravity. Our solutions contain two arbitrary integration constants in the blackening factor <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>f</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>r</a:mi> <a:mo stretchy="false">)</a:mo> </a:mrow> </a:math> , allowing for an extremal configuration. Solution I reproduces a previously reported AdS black hole when one of the integration constants in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>f</e:mi> <e:mo stretchy="false">(</e:mo> <e:mi>r</e:mi> <e:mo stretchy="false">)</e:mo> </e:math> vanishes. For our black hole configurations, the scalar curvature is constant and negative, corresponding to the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>AdS</i:mi> <i:mn>2</i:mn> </i:msub> </i:math> spacetime. In order to elucidate their black hole nature, we explore the causal structure of these solutions with the aid of suitable Kruskal-like coordinates and Penrose diagrams. By employing the Hamilton-Jacobi method, we construct a boundary counterterm that renders a renormalized action with a vanishing variation. We use this finite action for the partition function in the semiclassical approximation. We establish a consistent thermodynamics, verified by the first law, for our black hole solutions, including the extremal case. Finally, we perform a holographic analysis of the effective theory at the boundary of the black hole solution I. This theory is characterized by a Schwarzian action supplemented by a black hole mass term determined by the two integration constants in <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>f</k:mi> <k:mo stretchy="false">(</k:mo> <k:mi>r</k:mi> <k:mo stretchy="false">)</k:mo> </k:math> . We also examine the holographic implications of the boundary counterterm.