  There is an essentially unique way to associate to any Riemann surface a
measure on its simple loops, such that the collection of measures satisfy a
strong conformal invariance property. Wendelin Werner constructed these random
simple loops as boundaries of Brownian loops, and so they correspond in the zoo
of statistical mechanics models to central charge $0$, or Schramm-Loewner
Evolution (SLE) parameter $\kappa=8/3$. The goal of this paper is to construct
a family of measures on simple loops on Riemann surfaces that satisfies a
conformal covariance property, and that would correspond to SLE parameter
$\kappa=2$ (central charge $-2$). On planar annuli, this loop measure was
already built by Adrien Kassel and Rick Kenyon. We will give an alternative
construction of this loop measure on planar annuli, investigate its conformal
covariance, and finally extend this measure to general Riemann surfaces. This
gives an example of a Malliavin-Kontsevich-Suhov loop measure in non-zero
central charge.
