Erlangen 2026 – scientific programme

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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik

MP 3: Quantum Field Theory I: Axiomatic, Probabilistic, Algebraic, Conformal

MP 3.2: Invited Talk

Tuesday, March 17, 2026, 16:45–17:15, KH 02.013

Schwarzian Field Theory for Probabilists — •Peter Wildemann — University of Geneva

What does Liouville field theory, the SYK random matrix model and JT quantum gravity have in common? If you'd ask a physicist in recent years, they would be quick to point out that the low-energy behaviour of all these models should be described by the Schwarzian field theory. In itself, the latter can be understood as a probability measure on a quotient of the group of circle-diffeomorphisms Diff(T)/PSL(2,R). We discuss a rigorous approach constructing the measure in terms of a non-linear transformation of Brownian bridges, following ideas by Belokurov--Shavgulidze. Furthermore, we present new results that uniquely characterise the measure in terms of an appropriate change-of-variables formula, which can be seen as an analogue of the Cameron--Martin theorem on the space of circle diffeomorphisms. As a byproduct, we also obtain a short proof for the calculation of the measure's partition function (i.e. total mass), confirming a prediction by Stanford--Witten. This talk is based on joint work with Roland Bauerschmidt and Ilya Losev.

Keywords: Schwarzian Field Theory; Probability