Probabilistic Definition of the Schwarzian Field Theory

We define the Schwarzian Field Theory as a finite measure on Diff¹(T)/PSL(2,R) and compute its generalised partition functions exactly using methods of stochastic analysis. The Schwarzian Field Theory has attracted recent attention due to its role in the analysis of the Sachdev--Ye--Kitaev model and as the proposed holographic dual to Jackiw--Teitelboim gravity. In two companion papers by Losev, the predicted exact cross-ratio correlation functions for non-crossing Wilson lines and the large deviations are derived from the probability measure.}