An elementary approach to quantum length of SLE

Abstract

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