Controlled Loewner–Kufarev Equation Embedded into the Universal Grassmannian

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.