Cutting $\gamma$-Liouville quantum gravity by Schramm-Loewner evolution for $\kappa \not\in \{\gamma^2, 16/\gamma^2\}$

Abstract

There are many deep and useful theorems relating Schramm-Loewner evolution (SLE$_\kappa$) and Liouville quantum gravity ($\gamma$-LQG) in the case when the parameters satisfy $\kappa \in \{\gamma^2, 16/\gamma^2\}$. Roughly speaking, these theorems say that the SLE$_\kappa$ curve cuts the $\gamma$-LQG surface into two or more independent $\gamma$-LQG surfaces. We extend these theorems to the case when $\kappa \not\in \{\gamma^2, 16/\gamma^2\}$. Roughly speaking we show that if we have an appropriate variant of SLE$_\kappa$ and an independent $\gamma$-LQG disk, then the SLE curve cuts the LQG disk into two or more $\gamma$-LQG surfaces which are conditionally independent given the values along the SLE curve of a certain collection of auxiliary imaginary geometry fields, viewed modulo conformal coordinate change. These fields are sampled independently from the SLE and the LQG and have the property that that the sum of the central charges associated with the SLE$_\kappa$ curve, the $\gamma$-LQG surface, and the auxiliary fields is 26. This condition on the central charge is natural from the perspective of bosonic string theory. We also prove analogous statements when the SLE curve is replaced by, e.g., an LQG metric ball or a Brownian motion path. Statements of this type were conjectured by Sheffield and are continuum analogs of certain Markov properties of random planar maps decorated by two or more statistical physics models. We include a substantial list of open problems.