Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian

TL;DR

This paper introduces controlled Loewner-Kufarev equations, explores their algebraic structure, and connects solutions to the Grassmannian, tau-functions, and integrable systems, providing explicit formulas involving the driving function's signature.

Contribution

It develops a novel class of controlled Loewner-Kufarev equations and links their solutions to the Grassmannian and algebraic structures, offering explicit formulas in terms of the driving function.

Findings

Solutions expressed via the signature of the driving function.

Connection between the equations and the Grassmannian structure.

Relation of the Grunsky matrix with integrable systems.

Abstract

We introduce the class of controlled Loewner-Kufarev equations and consider aspects of their algebraic nature. We lift the solution of such a controlled equation to the (Sato)-Segal-Wilson Grassmannian, and discuss its relation with the tau-function. We briefly highlight relations of the Grunsky matrix with integrable systems and conformal field theory. Our main result is the explicit formula which expresses the solution of the controlled equation in terms of the signature of the driving function through the action of words in generators of the Witt algebra.