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.
Highlights
Negatively curved backgrounds, including a sum over higher genus topologies in the bulk
This is properly regarded as a time-dependent background in two dimensional string theory, rather than the minimal string4 and this has a dual description by considering a time-dependent background of the well-known c = 1 matrix quantum mechanics [24, 46, 47] as found in [40]
We argued an equivalence between the JT gravity on an anti de-Sitter space and the two dimensional string theory defined by a time-like Liouville CFT coupled to the space-like Liouville one
Summary
We study the partition function of JT gravity on a fixed background. We are interested in the hyperbolic disk and double-trumpet geometries. We write the action of JT gravity on a two dimensional manifold M as follows:. Where γ and K are the induced metric and trace of the extrinsic curvature on the boundary ∂M. One can add the Einstein-Hilbert action to the above action, but it is just topological and gives a constant contribution −S0χ(M ) to the on-shell action, where S0 is a coefficient of the Einstein-Hilbert action and χ(M ) is the Euler characteristic of the. For a fixed background geometry as we will consider in the rest of the paper, this topological suppression factor does not play any important role. We will omit the EinsteinHilbert action in the following
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