Probability and Mathematical Physics Seminar

Beginning with the seminal work of Sheffield, there have been many deep and useful theorems relating Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG) when their parameters are matched, meaning $\kappa \in \{ \gamma^2, 16/\gamma^2\}$. Roughly speaking, the SLE curve cuts the LQG surface into two or more independent LQG surfaces. We extend these theorems to the setting of mismatched parameters: an LQG disk is cut by an SLE curve into two or more LQG surfaces which are conditionally independent given the values along the SLE curve of a certain collection of auxiliary fields. These fields are sampled independently of the LQG and SLE, and have the property that the central charges of the LQG, SLE and auxiliary fields sum to 26. This central charge condition is natural from the perspective of bosonic string theory. Similar statements hold when the SLE curve is replaced by, e.g., an LQG metric ball or a Brownian motion path. These statements are continuum analogs of certain Markov properties of random planar maps decorated by two or more statistical physics models.