An elementary approach to quantum length of SLE

Abstract

We present an elementary proof establishing the equality of the right and left-sided $\sqrt{\kappa}$-quantum lengths for an SLE$_\kappa$ curve, where $\kappa\in (0,4]$. We achieve this by demonstrating that the$\sqrt{\kappa}$-quantum length is equal to the $(\sqrt{\kappa}/2)$-Gaussian multiplicative chaos with reference measure given by half the conformal Minkowski content of the curve, multiplied by $2/(4-\kappa)$ for $\kappa\in (0,4)$ and by $1$ for $\kappa=4$. Our proof relies on a novel "one-sided" approximation of the conformal Minkowski content, which is compatible with the conformal change of coordinates formula.