Probabilistic Definition of the Schwarzian Field Theory

We provide mathematical foundations for the Schwarzian Field Theory as a finite Borel measure on $\mathrm{Diff}^1(\mathbb{T})/\mathrm{PSL}(2,\mathbb{R})$, a quotient of the space of circle reparametrisations. The measure is defined by a natural change of variables formula, which we show uniquely characterises it. We further compute its partition function (total mass) from this change of variable formula. The existence of the measure then follows from an explicit construction involving a nonlinear transformation of a Brownian Bridge, proposed by Belokurov--Shavgulidze. 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 this measure.