  In this notes we shall describe the relation of a certain class of simple
random curves arising in 2D statistical mechanics models in the scaling limit,
which can be described dynamically by Stochastic L{\oe}wner Evolutions (SLE),
and the equivalent Renormalisation Group (RG) theoretic interpretation in
Conformal Field Theory, as a fixed point of the RG flow. Further, we shall
recall the relation of this random curves with String Theory, and how one can
derive a general measure on such random paths, by using weighted regularised
determinants, which come from sections of twisted line bundles. Importantly,
the null vector at level two in the Verma module for the highest-weight
representation of the Virasoro algebra corresponds to a generalised Doob-Getoor
h-transform.
