We formulate the path integral for Jackiw-Teitelboim gravity in the second order formalism working directly with the metric and the dilaton in both Anti-de Sitter(AdS) and de Sitter space(dS). In the dS case with disk topology, the path integral evaluates the wavefunction of the universe which arises in the no-boundary proposal. We will also analyze the behavior in the presence of conformal matter. In the connected geometry with two boundaries, for both AdS and dS, we find that bosonic matter gives rise to a diverging contribution in the moduli space integral rendering the path integral ill-defined and shall discuss how this can be avoided. For the pure dS JT theory without matter, the path integral gives rise in general to the Hartle-Hawking wave function which describes an arbitrary number of disconnected universes produced by tunnelling "from nothing", or to transition amplitudes which describe the tunnelling of an initial state consisting of several contracting universes to a final state of several expanding universes. These processes can be described by a hologram consisting of Random Matrix Theory (RMT) or, after some modification on the gravity side, by a hologram with the RMT being replaced by SYK theory.
This talk is based on 2101.00596 and 2202.03130
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