Holography of the Loewner Energy

by Prof. Yilin Wang (IHES)

Monday Nov 7, 2022, 4:00 PM → 5:15 PM Europe/Paris

Centre de conférences Marilyn et James Simons (IHES)

The link between the hyperbolic geometry of 3-manifolds and the conformal metrics on their boundary has been explored extensively in the context of hyperbolic geometry and is also motivated by the AdS₃/CFT₂ correspondence. An elementary observation is that the group of Möbius transformations on the Riemann sphere coincides with the isometries of the hyperbolic 3-space H³. The Loewner energy is a Möbius-invariant quantity that measures the roundness of Jordan curves. It arises from large deviations of SLE0+ and is a Kähler potential on the universal Teichmüller space endowed with the Weil-Petersson metric. We show that the Loewner energy of a Jordan curve in the Riemann sphere equals the renormalized volume of a submanifold of H³ constructed using the Epstein surfaces associated with the hyperbolic metric on both sides of the curve. This is work in progress with Martin Bridgeman (Boston College), Ken Bromberg (Utah), and Franco Vargas-Pallete (Yale).