Yilin Wang

Holography of the Loewner energy

The Loewner energy is a conformally invariant quantity that measures the roundness of Jordan curves on the Riemann sphere. It is the action functional of the family of random curves SLE (arising as interfaces in critical 2D lattice models). Motivated by AdS/CFT correspondence and the fact that the conformal group of the Riemann sphere coincides with the isometry group of the hyperbolic 3-space H^3, we look for quantities defined geometrically in H^3 which equal the Loewner energy of a curve in the conformal boundary. We show that the Loewner energy equals the renormalized volume (closely related to the Einstein-Hilbert action) of a submanifold of H^3 constructed using the Epstein surfaces. I will review the basics of SLE, Loewner energy, and then discuss the holography of the Loewner energy.