A Renormalisation Group approach to Stochastic L{\oe}wner Evolutions and the Doob h-transform

Abstract

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.