Thermodynamics and statistical mechanics of induced Liouville gravity

In this paper we describe Liouville gravity which is induced by a set of quantum fields (constituents) and represents a two-dimensional analogue of Sakharov’s induced gravity. The important feature of the considered theory is the presence of massless constituents which are responsible for the appearance of the induced Liouville field. The role of the massive constituents is only to induce the cosmological constant. We consider the instanton solutions of the Euclidean Liouville gravity with negative and zero cosmological constants, some instantons being interpreted as two-dimensional anti–de Sitter $\left({\mathrm{AdS}}_2\right)$ black holes. We study the thermodynamics of all the solutions and conclude that their entropy is completely determined by the statistical-mechanical entropy of the massless constituents. This shows, in particular, that the constituents of the induced gravity are the true degrees of freedom of ${\mathrm{AdS}}_2$ black holes. Special attention is also paid to the induced Liouville gravity with a zero cosmological constant on a torus. We demonstrate the equivalence of its thermodynamics to the thermodynamics of BTZ black holes and comment on computations of the BTZ black hole entropy.