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

TL;DR

This paper explores the connection between Stochastic Loewner Evolutions, Renormalisation Group theory, and Conformal Field Theory, highlighting their relation to string theory and measures on random paths via advanced mathematical constructs.

Contribution

It provides a novel interpretation of SLE in terms of RG fixed points and links it to string theory and the Doob h-transform using twisted line bundles and Virasoro algebra.

Findings

SLE curves correspond to RG fixed points in CFT.

A measure on random paths is derived using twisted line bundles.

The Doob h-transform is related to null vectors in Virasoro modules.

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.