Flat JT Gravity and the BMS-Schwarzian

We consider Minkowskian Jackiw-Teitelboim (JT) gravity in Bondi gauge at finite temperature, with non-zero vacuum energy. Its asymptotic symmetries span an extension of the warped Virasoro group, dubbed "BMS$_2$", which we investigate in detail. In particular, we show that this extension has a single coadjoint orbit when central charges are real and non-zero. The ensuing BMS-Schwarzian action has no saddle points, and only coincides with the boundary action functional of flat JT gravity up to a crucial dilatonic zero-mode that ensures the existence of a well-defined bulk variational principle. We evaluate the corresponding gravitational partition function, which turns out to be one-loop exact precisely thanks to the presence of such a zero-mode.