The complex Liouville string: the gravitational path integral

We give a rigorous definition of sine dilaton gravity in terms of the worldsheet theory of the complex Liouville string arXiv:2409.17246. The latter has a known exact solution that we leverage to explore the gravitational path integral of sine dilaton gravity - a quantum deformation of dS JT gravity that admits both AdS$_2$ and dS$_2$ vacua. We uncover that the gravitational path integral receives contributions from new saddles describing transitions between vacua in a third-quantized picture. We also discuss the sphere and disk partition function in this context and contrast our findings with other recent work on this theory.