One-dimensional Quantum Gravity and the Schwarzian theory

We develop a model of one-dimensional (Conformal) Quantum Gravity. By discussing the connection between Goldstone and Gauge theories, we establish that this effectively computes partition function Schwarzian theory where $\textrm{SL}(2,\mathbb{R})$ symmetry is realized on base space. The computation straightforward, involves local quantum measure does not rely localization arguments. Non-localities in are exclusively related to value fixed gauge invariant moduli. Furthermore, study properties these models when all degrees freedom allowed fluctuate. discuss UV finiteness systems emergence Planck's length.