Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 108, 25 pages      arXiv:1809.00534      https://doi.org/10.3842/SIGMA.2020.108

Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian

Takafumi Amaba a and Roland Friedrich b
a) Fukuoka University, 8-19-1 Nanakuma, Jônan-ku, Fukuoka, 814-0180, Japan
b) ETH Zürich, D-GESS, CH-8092 Zurich, Switzerland

Received June 30, 2020, in final form October 22, 2020; Published online October 29, 2020

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.

Key words: Loewner-Kufarev equation; Grassmannian; conformal field theory; Witt algebra; free probability theory; Faber polynomial; Grunsky coefficient; signature.

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