Conformal field theory and Loewner-Kufarev evolution

TL;DR

This paper surveys recent advances in understanding contour dynamics through conformal field theory, highlighting the connections between algebraic structures like the Virasoro algebra and classical and stochastic Loewner evolutions.

Contribution

It elucidates the relationship between the Virasoro algebra and contour dynamics, bridging conformal field theory with Loewner-Kufarev evolution.

Findings

Virasoro algebra plays a key role in contour dynamics.

Connections established between CFT and Loewner evolutions.

Survey of recent progress in the field.

Abstract

One of the important aspects in recent trends in complex analysis has been the increasing degree of cross-fertilization between the latter and mathematical physics with great benefits to both subjects. Contour dynamics in the complex plane turned to be a meeting point for complex analysts, specialists in stochastic processes, and mathematical physicists. This was stimulated, first of all, by recent progress in understanding structures in the classical and stochastic L\"owner evolutions, and in the Laplacian growth. The Virasoro algebra provides a basic algebraic object in conformal field theory (CFT) so it was not surprising that it turned to play an important role of a structural skeleton for contour dynamics. The present paper is a survey of recent progress in the study of the CFT viewpoint on contour dynamics, in particular, we show how the Witt and Virasoro algebras are related with the stochastic L\"owner and classical L\"owner-Kufarev equations.