Epstein curves and holography of the Schwarzian action

Vortrag von Dr. Catherine Wolfram

Datum: 12.11.25  Zeit: 17.15 - 18.45  Raum: ETH HG G 43

The circle can be seen as the boundary at infinity of the hyperbolic plane. We give a 1-to-2 dimensional holographic interpretation of the Schwarzian action, by showing that the Schwarzian action (which is a function of a diffeomorphism of the circle) is equal to the hyperbolic area enclosed by an "Epstein curve" in the disk. A dimension higher, the Epstein construction was used to relate the Loewner energy (a function of a Jordan curve related to SLE and Brownian loop measures) to renormalized volume in hyperbolic 3-space.

In this talk I will explain how to construct the Epstein curve, how the bi-local observables of Schwarzian field theory can be interpreted as a renormalized hyperbolic length using the same Epstein construction, and (time permitting) discuss a bit what we know so far about the relationship between the Schwarzian action and the Loewner energy. This is based on joint work with Franco Vargas Pallete and Yilin Wang.