JT gravity limit of Liouville CFT and matrix model

Abstract

In this paper we study a connection between Jackiw-Teitelboim (JT) gravity on two-dimensional anti de-Sitter spaces and a semiclassical limit of c < 1 two-dimensional string theory. The world-sheet theory of the latter consists of a space-like Liouville CFT coupled to a non-rational CFT defined by a time-like Liouville CFT. We show that their actions, disk partition functions and annulus amplitudes perfectly agree with each other, where the presence of boundary terms plays a crucial role. We also reproduce the boundary Schwarzian theory from the Liouville theory description. Then, we identify a matrix model dual of our two-dimensional string theory with a specific time-dependent background in c = 1 matrix quantum mechanics. Finally, we also explain the corresponding relation for the two-dimensional de-Sitter JT gravity.