It is a general problem to connect the path integral point of view to quantum field theory with exact solutions which are often obtained by algebraic methods that are difficult to rigorously relate to the path integral. I will discuss two examples, the Schwarzian Field Theory and fractional correlations of the Sine-Gordon Field Theory at the free fermion point, where the path integral leads to exact solutions. In the probabilistic definition of the Schwarzian Theory, the partition function and cross-ratio correlation functions can be computed, confirming results of Stanford--Witten and of Mertens-Turiaci-Verlinde. In the Sine-Gordon Theory, we compute the fractional correlation functions, confirming predictions of Bernard--Leclaire and Lukyanov--Zamolodchikov (at the free fermion point).

This talk is based on joint works with and by Ilya Losev and Peter Wildemann for the Schwarzian Field Theory and with Scott Mason and Christian Webb for the Sine-Gordon Field Theory.