On JT gravity path integrals and the tunneling process

The Jackiw-Teitelboim (JT) gravity path integral of the trumpet can be interpreted as a transition amplitude from an older black hole to a younger one, accompanied by the emission of a baby universe, represented by the geodesic boundary of the trumpet. However, this interpretation becomes less straightforward for geometries with higher genus and multiple geodesic boundaries. In this paper, we examine the path integral for these more complex geometries and find that maintaining this interpretation requires accounting for a portion of the moduli space.